Answer:
The volume of the solid is [tex]\frac{48\pi}{5}[/tex]
Step-by-step explanation:
Consider the provided information.
The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola [tex]x=\sqrt6y^2[/tex]
Therefore, diameter is [tex]d=\sqrt6y^2[/tex]
Radius will be [tex]r=\frac{\sqrt6y^2}{2}[/tex]
We can calculate the area of circular disk as: πr²
Substitute the respective values we get:
[tex]A=\pi(\frac{\sqrt6y^2}{2})^2[/tex]
[tex]A=\pi(\frac{6y^4}{4})=\frac{3\pi y^4}{2}[/tex]
Thus the volume of the solid is:
[tex]V=\int\limits^2_0 {\frac{3\pi y^4}{2}} \, dy[/tex]
[tex]V=[{\frac{3\pi y^5}{2\times 5}}]^2_0[/tex]
[tex]V=\frac{48\pi}{5}[/tex]
Hence, the volume of the solid is [tex]\frac{48\pi}{5}[/tex]
The volume of solid represent the how much space an object occupied. In the given problem volume can be determine by taking the integration of Area of solid.
The volume of solid is [tex]\frac{48\pi }{5}[/tex].
Given:
The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola is [tex]x=\sqrt{6}y^2[/tex].
The diameter of the solid is [tex]d=\sqrt{6}y^2[/tex].
Calculate the radius of the solid.
[tex]r=\frac{d}{2}\\r=\frac{\sqrt{6}y^2}{2}[/tex]
Write the expression for area of circular disk.
[tex]A=\pi r^2\\A=\pi (\frac{\sqrt{6}y^2}{2})^2\\A=\frac{3\pi y^4}{2}[/tex]
Calculate the volume of solid.
[tex]V=\int\limits^2_0 {\frac{3\pi y^4 }{2} } \, dy\\V=[\frac{3\pi y^5}{2\times 5}]_{0}^{2}\\V=\frac{48\pi }{5}[/tex]
Thus, the volume of solid is [tex]\frac{48\pi }{5}[/tex] .
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The set of integers is closed under the operation of addition.
A: Which equation illustrates this concept?
B: Which statement correctly explains this concept?
Select one answer for question A and one answer for question B.
A: 2+27=29
A: 34÷4=172
A: 1−3=−2
A: 2⋅6=12
B: The sum of the integers 2 and 27 is the integer 29, which demonstrates that integers are closed under addition.
B: The quotient of the integers 34 and 4 is the integer 172, which demonstrates that integers are closed under addition.
B: The difference of the integers 1 and 3 is not an integer, −2, which does not demonstrate that integers are closed under addition.
B: The product of the integers 2 and 6 is not an integer, 12, which does not demonstrate that integers are closed under addition.
Answer:
A: 2+27=29
B: The sum of the integers 2 and 27 is the integer 29, which demonstrates that integers are closed under addition.
Step-by-step explanation:
Since, closed property of addition for a set A is defined as,
∀ x, y ∈ A ⇒ x + y ∈ A,
∵ Set of integer is closed under multiplication,
If Z represents the set of integer,
Then 2, 27 ∈ Z ⇒ 2 + 27 = 29 ∈ Z,
Hence, the equation illustrates given statement,
2+27 = 29
The statement that correctly explains given statement,
The sum of the integers 2 and 27 is the integer 29, which demonstrates that integers are closed under addition.c
Farmer bob's square plot ofland is slowly eroding away. Worried about the future of his farm. Farmer Bob measures the rate of erosion and finds that the length of each side of his square plot is decreasing at the constant rate of 2 feet/year. If he currently owns 250,000 square feet of land, what is the current rate of change of the area of Farmer Bob's land?
a) Farmer bob is losing 2,000 square feet of land per year
b) losing 1,000,000 square feet of land per year
c) losing 1,000 square feet of land per year
d) losing 4 square feet of land per year
Answer:
Option A.
Step-by-step explanation:
Area of a square is
[tex]A=x^2[/tex] .... (1)
where, x is side length.
The length of each side of his square plot is decreasing at the constant rate of 2 feet/year.
[tex]\dfrac{dx}{dt}=2[/tex]
It is given that bob currently owns 250,000 square feet of land.
Fist find the length of each side.
[tex]A=250000[/tex]
[tex]x^2=250000[/tex]
Taking square root on both sides.
[tex]x=500[/tex]
Differentiate with respect to t.
[tex]\dfrac{dA}{dt}=2x\dfrac{dx}{dt}[/tex]
Substitute x=500 and [tex]\frac{dx}{dt}=2[/tex] in the above equation.
[tex]\dfrac{dA}{dt}=2(500)(2)[/tex]
[tex]\dfrac{dA}{dt}=2000[/tex]
Farmer bob is losing 2,000 square feet of land per year.
Therefore, the correct option is A.
A certain car depreciates such that its value at the end of each year is p % less than its value at the end of the previous year. If that car was worth a dollars on December 31, 2010 and was worth b dollars on December 31, 2011, what was the car worth on December 31, 2013 in terms of a and b ?
Answer:
b(b/a)^2
Step-by-step explanation:
Given that the value of the car depreciates such that its value at the end of each year is p % less than its value at the end of the previous year and that car was worth a dollars on December 31, 2010 and was worth b dollars on December 31, 2011, then
b = a - (p% × a) = a(1-p%)
b/a = 1 - p%
p% = 1 - b/a = (a-b)/a
Let the worth of the car on December 31, 2012 be c
then
c = b - (b × p%) = b(1-p%)
Let the worth of the car on December 31, 2013 be d
then
d = c - (c × p%)
d = c(1-p%)
d = b(1-p%)(1-p%)
d = b(1-p%)^2
d = b(1- (a-b)/a)^2
d = b((a-a+b)/a)^2
d = b(b/a)^2 = b^3/a^2
The car's worth on December 31, 2013 = b(b/a)^2 = b^3/a^2
21. The parent function of the following graph is f(x) = 2^x. What is the equation of the following graph?
Answer:
5
Step-by-step explanation:
Plug x = 0 into f(x) to get
f(x) = 2^x
f(0) = 2^0
f(0) = 1
The y intercept (0,1) is on the graph of f(x).
The y intercept for the red curve shown is (0,3). It has been moved up two units compared to (0,1)
Therefore, g(x) = f(x)+2 where g(x) represents the red curve.
g(x) = f(x) + 2
[tex]g(x) = 2^x + 2[/tex] is the answer
Paper depot is swnding out 28 trucks to deliver paper to customers today. Each truck is being loaded with 3 cases of paper. 283 of all the cases are plain white paper. How many cases of all types of paper will be delivered today?
Answer:
84 cases
Step-by-step explanation:
Given that:
Number of trucks: 28
Paper cases each truck can load = 3
Total cases of white paper = 283
So the cases delivered will be = 28 *3 = 84 cases will be delivered today
i hope it will help you!
Mary scored a total of 27 points in a basketball game. During the game, all her points came from 3-point shots and 2-point shots. The number of 3-point shots she made is 4 more than her 2-point shots. How many 3-point shots and 2-point shots did she make?
A) 2 three-point shots and 6 two-point shots
B) 3 three-point shots and 7 two-point shots
C) 6 three=point shots and 2 two-point shots
D) 7 three-point shots and 3 two-point shots
Answer:
D
Step-by-step explanation:
7x3=21 and 3x2=6
21+6=27
D) The number of three point shots is 7 and the number of two points shots is 3.
What are Linear Equations?Linear equations are equation involving one or more expressions including variables and constants and the variables are having no exponents or the exponent of the variable is 1.
Given that,
Mary scored a total of 27 points in a basketball game, all from 3-point shots and 2-point shots.
Let x be the number of 3-point shots and y be the number of 2-point shots.
3x + 2y = 27 [Equation 1]
The number of 3-point shots she made is 4 more than her 2-point shots.
x = y + 4 [Equation 2]
Substitute [Equation 2] in [Equation 1].
3(y + 4) + 2y = 27
3y + 12 + 2y = 27
5y = 15
y = 3
x = y + 4 = 3 + 4 = 7
Hence the option is D) 7 three-point shots and 3 two-point shots.
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Please Help!!
Create two radical equations: one that has an extraneous solution, and one that does not have an extraneous solution. Use the equation below as a model:
[tex]a\sqrt{x+b} +c=d[/tex]
Given
[tex]a\sqrt{x+b}+c=d[/tex]
we have
[tex]\sqrt{x+b}=\dfrac{d-c}{a}[/tex]
Squaring both sides, we have
[tex]x+b=\dfrac{(d-c)^2}{a^2}[/tex]
And finally
[tex]x=\dfrac{(d-c)^2}{a^2}-b[/tex]
Note that, when we square both sides, we have to assume that
[tex]\dfrac{d-c}{a}>0[/tex]
because we're assuming that this fraction equals a square root, which is positive.
So, if that fraction is positive you'll actually have roots: choose
[tex]a=1,\ b=0,\ c=2,\ d=6[/tex]
and you'll have
[tex]\sqrt{x}+2=6 \iff \sqrt{x}=4 \iff x=16[/tex]
Which is a valid solution. If, instead, the fraction is negative, you'll have extraneous roots: choose
[tex]a=1,\ b=0,\ c=10,\ d=4[/tex]
and you'll have
[tex]\sqrt{x}+10=4 \iff \sqrt{x}=-6[/tex]
Squaring both sides (and here's the mistake!!) you'd have
[tex]x=36[/tex]
which is not a solution for the equation, if we plug it in we have
[tex]\sqrt{x}+10=4 \implies \sqrt{36}+10=4 \implies 6+10=4[/tex]
Which is clearly false
Which graph represents a quadratic function with a vertex at (0, 0)? On a coordinate plane, a parabola opens up. It goes through (negative 5, 6), has a vertex of (0, 1), and goes through (5, 6). On a coordinate plane, a parabola opens up. It goes through (negative 6, 6), has a vertex of (0, negative 1), and goes through (6, 6). On a coordinate plane, a parabola opens up. It goes through (negative 5, 6), has a vertex of (0, 0), and goes through (5, 6). On a coordinate plane, a parabola opens up. It goes through (negative 2.5, 6), has a vertex of (3, 0), and goes through (7.5, 4).
Answer:
On a coordinate plane, a parabola opens up. It goes through (negative 5, 6), has a vertex of (0, 0), and goes through (5, 6).
Step-by-step explanation:
Since you want a graph with a vertex of (0, 0), choose the one that is described as having a vertex of (0, 0).
Answer:
Option C.
Step-by-step explanation:
We need to find the graph which represents a quadratic function with vertex at (0,0).
The graph of quadratic function is a parabola (either upward or downward) and the extreme point of the parabola is know as vertex.
All graphs represent different parabolas.
Vertex of first parabola = (0,1)
Vertex of second parabola = (0,-1)
Vertex of third parabola = (0,0)
Vertex of fourth parabola = (3,0)
In option C, a parabola opens up on a coordinate plane. It goes through (-5, 6), has a vertex of (0, 0), and goes through (5, 6).
Only third graph represents a quadratic function with a vertex at (0, 0).
Therefore, the correct option is C.
A paint manufacturer uses a machine to fill gallon cans with paint (1 galequals128 ounces). The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.5 ounce. Assume the population of volumes is normally distributed.(a) Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 0.80 ounce.(b) The sample mean is 127 ounces. With a sample size of 8, a 90% level of confidence, and a population standard deviation of 0.80 ounce, does it seem possible that the population mean could be exactly 128 ounces? Explain.
Answer:
a) To determine the minimum sample size we need to use the formula shown in the picture 1.
E is the margin of error, which is the distance from the limits to the middle (the mean) of the confidence interval. This means that we have to divide the range of the interval by 2 to find this distance.
E = 0.5/2 = 0.25
Now we apply the formula
n = (1.645*0.80/0.25)^2 = 27.7 = 28
The minimum sample size would be 28.
b) To answer the question we are going to make a 90% confidence interval. The formula is:
(μ - E, μ + E)
μ is the mean which is 127. The formula for E is shown in the picture.
E = 0.80*1.645/√8 = 0.47
(126.5, 127.5)
This means that the true mean is going to be contained in this interval 90% of the time. This is why it doesn't seem possible that the population mean is exactly 128.
(a) Minimum sample size needed for a 90% confidence interval is 7.
(b) With a sample mean of 127 ounces, 128 ounces seems unlikely for the population mean.
To solve this problem, we can use the formula for the confidence interval of the population mean:
[tex]\[ \text{Confidence Interval} = \text{Sample Mean} \pm Z \left( \frac{\text{Population Standard Deviation}}{\sqrt{\text{Sample Size}}} \right) \][/tex]
Where:
- Sample Mean = 127 ounces
- Population Standard Deviation = 0.80 ounce
- Z = Z-score corresponding to the desired confidence level
- Sample Size = n
(a) To determine the minimum sample size required for a 90% confidence interval:
We first need to find the Z-score corresponding to a 90% confidence level. We'll use a Z-table or a calculator. For a 90% confidence level, the Z-score is approximately 1.645.
[tex]\[ \text{Margin of Error} = Z \left( \frac{\text{Population Standard Deviation}}{\sqrt{\text{Sample Size}}} \right) \][/tex]
Given that the margin of error is 0.5 ounce, we can rearrange the formula to solve for the sample size:
[tex]\[ 0.5 = 1.645 \left( \frac{0.80}{\sqrt{n}} \right) \][/tex]
Solving for ( n ):
[tex]\[ \sqrt{n} = \frac{1.645 \times 0.80}{0.5} \][/tex]
[tex]\[ \sqrt{n} = 2.632 \][/tex]
[tex]\[ n = (2.632)^2 \][/tex]
[tex]\[ n \approx 6.92 \][/tex]
Since the sample size must be a whole number, we round up to the nearest whole number. Therefore, the minimum sample size required is 7.
(b) To determine if it's possible that the population mean could be exactly 128 ounces with a sample mean of 127 ounces, a sample size of 8, and a 90% confidence level:
[tex]\[ \text{Confidence Interval} = 127 \pm 1.645 \left( \frac{0.80}{\sqrt{8}} \right) \][/tex]
[tex]\[ \text{Confidence Interval} = 127 \pm 1.645 \left( \frac{0.80}{\sqrt{8}} \right) \][/tex]
[tex]\[ \text{Confidence Interval} = 127 \pm 1.645 \left( \frac{0.80}{2.828} \right) \][/tex]
[tex]\[ \text{Confidence Interval} = 127 \pm 1.645 \times 0.283 \][/tex]
[tex]\[ \text{Confidence Interval} = 127 \pm 0.466 \][/tex]
The confidence interval is ( (126.534, 127.466) ).
Since 128 ounces is not within the confidence interval, it seems unlikely that the population mean could be exactly 128 ounces.
Due to a manufacturing error, four cans of regular soda were accidentally filled with diet soda and placed into an 18 pack. Suppose that two cans are randomly selected from the 18 pack.a. Determine the probability that both contain diet soda. (Round to four decimal places as needed.)b. Determine the probability that both contain regular soda. (Round to four decimal places as needed.)Would this be unusual?A. YesB. Noc. Determine the probability that exactly one is diet and exactly one is regular. (Round to four decimal places as needed.)
The probability of drawing two diet sodas is 0.0392, two regular sodas is 0.5948, and one of each is 0.3660. It would be unusual to select two diet sodas.
Explanation:To answer this question, we will be using combinatorial probability. The pack contains 4 cans of diet soda and 14 cans of regular soda (18 in total).
a. The probability that both cans are diet soda can be calculated as follows: There are 4 ways to choose the first can of diet soda and 3 ways to choose the second one. Thus, there are 4 * 3 = 12 favorable outcomes. There are 18 ways to choose the first can and 17 ways to choose the second, totaling 18 * 17 = 306 possible outcomes. Hence, the probability is 12/306 = 0.0392b. The probability that both cans are regular soda can be calculated similarly: There are 14 ways to select the first can of regular soda and 13 ways to select the second one. So, there are 14 * 13 = 182 favorable outcomes. Using the same total possible outcomes, the probability is 182/306 = 0.5948c. The probability that one is a diet soda and one is a regular soda can also be calculated: There are 4 ways to select the diet soda and 14 ways to select the regular soda. Thus, there are 4 * 14 = 56 favorable outcomes. However, since the soda can be selected in any order (regular then diet or diet then regular), we double these outcomes, resulting in 112. Hence, the probability is 112/306 = 0.3660
Unusual results are typically those that have low probability. So in this context, it would be unusual to select two diet sodas (a).
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The probabilities of selecting two cans of diet soda, regular soda, and one of each (diest and regular) from an 18 pack are 0.0235, 0.5378, and 0.3878, respectively.
Explanation:These types of calculations fall under the category of combinatorics, specifically combinations. We are interested in the number of ways we can select cans from a total of 18 where order does not matter.
a. The probability that both randomly selected cans are diet soda is calculated by the formula: (Number of ways to select diet soda) / (Total ways to select two cans). Here we have 4 cases in which we could select diet soda and 18 ways to select any two cans from the pack. Hence, we calculate the probability as:
(4/18) * (3/17) = 0.0235
b. In a similar manner, the probability that both randomly selected cans are non-diet soda (regular soda) is calculated as:
(14/18) * (13/17) = 0.5378
These results are not unusual as there are more regular soda cans in the pack, hence the probability of picking two regular soda cans is higher.
c. The probability that exactly one is diet and exactly one is regular, we have two cases: selecting diet soda first and then regular soda second or selecting regular soda first then diet soda. Hence we calculate the probability as:
(4/18) * (14/17) + (14/18) * (4/17) = 0.3878
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Two bicycles are traveling along perpendicular roads. Bicycle A is traveling due east at 4 mi/hr, and bicycle B is travelling due north at 6 mi/hr. At noon, when bicycle A reaches the intersection, bicycle B is 9 mi away and moving toward the same intersection If t is the number of hours after noon, the bicycles are closest together when t isA. 0B. 27/26C. 9/5D. 3/2E. 14/13
Answer:
Step-by-step explanation:
Given
speed of cyclist A is [tex]v_a=4 mi/hr[/tex]
speed of cyclist B is [tex]v_b=6 mi/hr[/tex]
At noon cyclist B is 9 mi away
after noon Cyclist B will travel a distance of 6 t and cyclist A travel 4 t miles in t hr
Now distance of cyclist B from intersection is 9-6t
Distance of cyclist A from intersection is 4 t
let distance between them be z
[tex]z^2=(9-6t)^2+(4t)^2[/tex]
Differentiate z w.r.t time
[tex]2z\frac{\mathrm{d} z}{\mathrm{d} t}=2\times (9-6t)\times (-6)+2\times (4t)\times 4[/tex]
[tex]z\frac{\mathrm{d} z}{\mathrm{d} t}=(-6)(9-6t)+4(4t)[/tex]
[tex]\frac{\mathrm{d} z}{\mathrm{d} t}=\frac{16t+36t-54}{z}[/tex]
Put [tex]\frac{\mathrm{d} z}{\mathrm{d} t}\ to\ get\ maximum\ value\ of\ z[/tex]
therefore [tex]52t-54=0[/tex]
[tex]t=\frac{54}{52}[/tex]
[tex]t=\frac{27}{26} hr [/tex]
Combine like terms to create an equivalent expression. 4\left(1.75y-3.5\right)+1.25y4(1.75y−3.5)+1.25y4, left parenthesis, 1, point, 75, y, minus, 3, point, 5, right parenthesis, plus, 1, point, 25, y
Answer:
8.25y-14
Step-by-step explanation:
The simplified expression is 8.25y - 14, which is obtained by combining the like terms 7y and 1.25y and adding the constant term -14.
How to combine like terms?Break down the expression step by step.
4(1.75y - 3.5): This is a product of a number and a sum. Distribute the 4 to get 4 × 1.75y + 4 × (-3.5).
4 × 1.75y: This is multiplication of a number and a variable. The product is 7y.
4 × (-3.5): This is multiplication of a number and a constant. The product is -14.
1.25y: This is a single term.
Now, combine the like terms. Like terms are terms that have the same variable and the same exponent. In this case, the like terms are 7y and 1.25y.
When combined, the like terms are, 7y + 1.25y = 8.25y.
Add the constant term, -14.
Therefore, the simplified expression is 8.25y - 14.
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Please help with these! I don't know how they work.
Answer:
1. all real numbers
2. y ≥ -8
3. It is all of the possible values of a function
4. Domain:{-4,-2,0,2,4} and Range:{-2,0,1,2,3}
Step-by-step explanation:
Let f:A→B be a function. In general sets A and B can be any arbitary non-empty sets.
Values in set A are the input values to the function and values in set B are the output values
Hence Set A is called the domain of the function f.
Set B is called co-domain or range of function f.
Now coming back to problem,
In first picture,
Given function is a straight line ⇒it can take any real number as its input
and for each value it gives a unique output value.
Hence output value is set of all real numbers, i.e. range of the function represented by the graph is set of all real numbers.
In the second picture,
The graph is x values are extending from -∞ to ∞ but the y values is the set of values of real numbers greater than -8 since we can see that the graph has global minimum of -8
Therefore range of the graph is y≥-8
in the third picture,
as we have already discussed the range of a function is the set of all possible output values of the function
In the fourth picture,
Let the function be 'f'.
from question we can tell that we can take only -4,-2,0,2,4 as the values for x and for corresponding x values we get 1,3,2,-2,0 as y values which are the output values.
hence we can tell that domain, which is set of input values, is {-4,-2,0,2,4}
and range, which is the of possible output values, is {-2,0,1,2,3}
The age of Noelle's dad is 6 less than 3 times Noelle's age. The sum of their ages is 74 . Find their ages. Provide your answer below: $$ Noelle: , Noelle's Dad
Answer:
The age of Noelle = 20 years
Age of Noelle's dad = 54 years
Step-by-step explanation:
Here, Let us assume:
The age of Noelle's age = m years
So, the age of Noelle's dad = 3 x ( Age of Noelle) - 6 = 3(m) - 6
Also, sum of both the ages = 74
So, sum of (Noelle's age + Noelle's dad's) age = 74 years
⇒ m + ( 3 m - 6) = 74
or, 4 m = 74+ 6 = 80
or,m = 80 / 4 = 20
⇒ m = 20
Hence, the age of Noelle = m - 3 = 20 years
Age of Noelle's dad = 3 m - 6 = 3(20) - 6 = 54 years
Two gardeners can do the weekly yard maintenance in 8 minutes if they work together. The older gardener takes 12 minutes more than the younger gardener to finish the job by himself. How long does it take for each gardener to do the weekly yard maintainence individually?
Answer:
Let x be the time taken( in minutes ) by younger gardener,
So, the one minute work of younger gardener = [tex]\frac{1}{x}[/tex]
Also, the time taken by older gardener = (x+12) minutes ( given ),
So, the one minute work of older gardener = [tex]\frac{1}{x+12}[/tex]
Total work done in one minute = [tex]\frac{1}{x}+\frac{1}{x+12}[/tex]
Now, total time taken = 8 minutes,
Total work done in one minute = [tex]\frac{1}{8}[/tex]
Thus,
[tex]\frac{1}{x}+\frac{1}{x+12}=\frac{1}{8}[/tex]
[tex]\frac{x+12+x}{x^2+12x}=\frac{1}{8}[/tex]
[tex]\frac{2x+12}{x^2+12x}=\frac{1}{8}[/tex]
[tex]16x + 96 = x^2+12x[/tex]
[tex]x^2 -4x -96=0[/tex]
[tex]x^2 - 12x + 8x - 96=0[/tex]
[tex]x(x-12) + 8(x-12)=0[/tex]
[tex](x+8)(x-12)=0[/tex]
By zero product product property,
x + 8 =0 or x - 12 =0
⇒ x = -8 ( not possible ), x = 12
Hence, the time taken by younger gardener = 12 minutes,
And, the time taken by older gardener = 12 + 12 = 24 minutes.
The Glee Club sold a total of 150 tickets to their spring concert. Student tickets cost $5.00 each and adult tickets cost $8.00 each. If they had $1,020 in ticket sales, how many adult tickets did they sell?
Answer: 60 student tickets were sold
90 adult tickets were sold
Step-by-step explanation:
Let x represent the total number of student tickets sold.
Let y represent the total number of adult tickets sold.
The Glee Club sold a total of 150 tickets to their spring concert. This means that
x + y = 150
x = 150 - y
Student tickets cost $5.00 each and adult tickets cost $8.00 each. If they had $1,020 in ticket sales,then,
5x + 8y = 1020 - - - - - -1
Substituting x = 150 - y into equation 1, it becomes
5(150 - y) + 8y = 1020
750 - 5y + 8y = 1020
- 5y + 8y = 1020 - 750
3y = 270
y = 270/3 = 90
x = 150 - 90 = 60
a. Consider the following algorithm segment: for i := 1 to 4, for j := 1 to i, [Statements in body of inner loop. None contain branching statements that lead outside the loop.] next j, next i. How many times will the inner loop be iterated when the algorithm is implemented and run?
b. Let n be a positive integer, and consider the following algorithm segment: for i := 1 to n, for j := 1 to i, [Statements in body of inner loop. None contain branching statements that lead outside the loop.], next j, next i. How many times will the inner loop be iterated when the algorithm is implemented and run?
The number of times the inner loop be iterated when the algorithm is implemented and run is 10.
We are given that;
i := 1 to 4, for j := 1
Now,
a. The inner loop will be iterated 10 times when the algorithm is implemented and run.
| i | j | Iterations |
|---|---|------------|
| 1 | 1 | 1 |
| 2 | 1 | 2 |
| 2 | 2 | 3 |
| 3 | 1 | 4 |
| 3 | 2 | 5 |
| 3 | 3 | 6 |
| 4 | 1 | 7 |
| 4 | 2 | 8 |
| 4 | 3 | 9 |
| 4 | 4 | 10 |
b. The number of iterations of the inner loop depends on the value of i. For each i, the inner loop runs from j = 1 to j = i.
So, the total number of iterations is the sum of i from i = 1 to i = n. This is a well-known arithmetic series, which can be written as:
[tex]$\sum_{i=1}^n i = \frac{n(n+1)}{2}$[/tex]
This is the formula for the number of iterations of the inner loop.
Therefore, by algorithm the answer will be 10.
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a. The inner loop will be iterated 10 times when the algorithm is implemented and run.
b. The total number of iterations for the inner loop when the algorithm is implemented and run will be (n^2 + n)/2.
a. In the given algorithm segment, we have two nested loops. The outer loop runs from 1 to 4, and the inner loop runs from 1 to i. Let's analyze how many times the inner loop will be iterated.
For i = 1, the inner loop will run once.
For i = 2, the inner loop will run twice.
For i = 3, the inner loop will run three times.
For i = 4, the inner loop will run four times.
Therefore, the total number of iterations for the inner loop can be calculated by summing the iterations for each value of i:
1 + 2 + 3 + 4 = 10
b. In this algorithm segment, we have the same nested loops as in part a, but the range of the outer loop is from 1 to n, where n is a positive integer.
The inner loop iterates from 1 to i, where i takes the values from 1 to n. So, for each value of i, the inner loop will run i times.
To determine the total number of iterations for the inner loop, we need to sum the iterations for each value of i from 1 to n:
1 + 2 + 3 + ... + n
This is an arithmetic series, and the sum of an arithmetic series can be calculated using the formula:
Sum = (n/2)(first term + last term)
In this case, the first term is 1, and the last term is n. Substituting these values into the formula, we get:
Sum = (n/2)(1 + n) = (n^2 + n)/2
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Help me!!! 100 points !!
Answer:
200000 times larger, its not in scientifc notation it should be 1.3*10^9, and the answer is 1.824*10^15,or182400000000000
Step-by-step explanation:
Answer:
Step-by-step explanation:
To find how many times larger, divide the CEO's salary by the teachers to get
5 000 000 / 25 000 = 200
this in scientific notation is 2.00 * 10^2
d - is not in scientific notation because there is more than one digit before the decimal place
1.3 * 10 ^9
e - 1.748 * 10^15
I need help with my math homework. The questions is: Find all solutions of the equation in the interval [0,2π).
√3 tan(x- π/8)-1=0
Answer:
[tex]\frac{7\pi}{24}[/tex] and [tex]\frac{31\pi}{24}[/tex]
Step-by-step explanation:
[tex]\sqrt{3} \tan(x-\frac{\pi}{8})-1=0[/tex]
Let's first isolate the trig function.
Add 1 one on both sides:
[tex]\sqrt{3} \tan(x-\frac{\pi}{8})=1[/tex]
Divide both sides by [tex]\sqrt{3}[/tex]:
[tex]\tan(x-\frac{\pi}{8})=\frac{1}{\sqrt{3}}[/tex]
Now recall [tex]\tan(u)=\frac{\sin(u)}{\cos(u)}[/tex].
[tex]\frac{1}{\sqrt{3}}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}[/tex]
or
[tex]\frac{1}{\sqrt{3}}=\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}[/tex]
The first ratio I have can be found using [tex]\frac{\pi}{6}[/tex] in the first rotation of the unit circle.
The second ratio I have can be found using [tex]\frac{7\pi}{6}[/tex] you can see this is on the same line as the [tex]\frac{\pi}{6}[/tex] so you could write [tex]\frac{7\pi}{6}[/tex] as [tex]\frac{\pi}{6}+\pi[/tex].
So this means the following:
[tex]\tan(x-\frac{\pi}{8})=\frac{1}{\sqrt{3}}[/tex]
is true when [tex]x-\frac{\pi}{8}=\frac{\pi}{6}+n \pi[/tex]
where [tex]n[/tex] is integer.
Integers are the set containing {..,-3,-2,-1,0,1,2,3,...}.
So now we have a linear equation to solve:
[tex]x-\frac{\pi}{8}=\frac{\pi}{6}+n \pi[/tex]
Add [tex]\frac{\pi}{8}[/tex] on both sides:
[tex]x=\frac{\pi}{6}+\frac{\pi}{8}+n \pi[/tex]
Find common denominator between the first two terms on the right.
That is 24.
[tex]x=\frac{4\pi}{24}+\frac{3\pi}{24}+n \pi[/tex]
[tex]x=\frac{7\pi}{24}+n \pi[/tex] (So this is for all the solutions.)
Now I just notice that it said find all the solutions in the interval [tex][0,2\pi)[/tex].
So if [tex]\sqrt{3} \tan(x-\frac{\pi}{8})-1=0[/tex] and we let [tex]u=x-\frac{\pi}{8}[/tex], then solving for [tex]x[/tex] gives us:
[tex]u+\frac{\pi}{8}=x[/tex] ( I just added [tex]\frac{\pi}{8}[/tex] on both sides.)
So recall [tex]0\le x<2\pi[/tex].
Then [tex]0 \le u+\frac{\pi}{8}<2 \pi[/tex].
Subtract [tex]\frac{\pi}{8}[/tex] on both sides:
[tex]-\frac{\pi}{8}\le u <2 \pi-\frac{\pi}{8}[/tex]
Simplify:
[tex]-\frac{\pi}{8}\le u <\pi (2-\frac{1}{8})[/tex]
[tex]-\frac{\pi}{8}\le u<\frac{15\pi}{8}[/tex]
So we want to find solutions to:
[tex]\tan(u)=\frac{1}{\sqrt{3}}[/tex] with the condition:
[tex]-\frac{\pi}{8}\le u<\frac{15\pi}{8}[/tex]
That's just at [tex]\frac{\pi}{6}[/tex] and [tex]\frac{7\pi}{6}[/tex]
So now adding [tex]\frac{\pi}{8}[/tex] to both gives us the solutions to:
[tex]\tan(x-\frac{\pi}{8})=\frac{1}{\sqrt{3}}[/tex] in the interval:
[tex]0\le x<2\pi[/tex].
The solutions we are looking for are:
[tex]\frac{\pi}{6}+\frac{\pi}{8}[/tex] and [tex]\frac{7\pi}{6}+\frac{\pi}{8}[/tex]
Let's simplifying:
[tex](\frac{1}{6}+\frac{1}{8})\pi[/tex] and [tex](\frac{7}{6}+\frac{1}{8})\pi[/tex]
[tex]\frac{7}{24}\pi[/tex] and [tex]\frac{31}{24}\pi[/tex]
[tex]\frac{7\pi}{24}[/tex] and [tex]\frac{31\pi}{24}[/tex]
The graph of function f a shown. Use the zeros and the turning points of the graph to find the rule for f.
Answer:
f(x) = x(x +4)(x -3)
Step-by-step explanation:
Zeros at -4, 0, and 3 tell you the factorization is ...
f(x) = a(x +4)(x)(x -3)
Then f(2) = a(6)(2)(-1) = -12a.
The graph shows f(2) = -12, so a=1. That makes the function rule:
f(x) = x(x +4)(x -3)
__
If you want it multiplied out, it will be
f(x) = x^3 +x^2 -12x
To find the rule for function f(x) using zeros and turning points, analyze the graph. In this case, with a horizontal line between 0 and 20, there are no zeros or turning points as the function doesn't cross the x-axis or change direction.
Explanation:To find the rule for a function f(x) using its zeros and turning points, we analyze the graphical representation of the function. If the graph is a horizontal line, such as when f(x) = 20 for all 0 ≤ x ≤ 20, the function does not have any zeros or turning points within that interval, as it does not cross the x-axis nor does it change direction. Considering this particular function, we conclude that the graph is indeed a horizontal line with no turning points or zeros between x=0 and x=20.
In the general process of graphing, to illustrate the change in f(x) as x varies, we plot specific (x,y) data pairs and use these to determine trends. For functions that are not constant, like the one described above, zeros are the x-values where the function crosses the x-axis (f(x) = 0), and turning points are found where the slope of the function changes sign, which can be determined by examining the first and second derivatives of the function. However, in this scenario, the horizontal nature of the graph precludes the presence of such features.
Henry was buying meals for his family. He bought 3 meals that included a chicken, corn, and a drink and spent a total of $20.25. He paid 6% sales tax. How much did each meal cost before tax?
Answer:
The price of each meal cost before tax is $ 6.367
Step-by-step explanation:
Given as :
The total price of three meals = $ 20.25
The sales tax included in the total price = 6 %
So, Let the cost of meal before sales tax = x
Or, x + 6 % of x = $ 20.25
or, x + 0.06 x = $ 20.25
Or, 1.06 x = $ 20.25
∴ x = [tex]\frac{20.25}{1.06}[/tex]
I.e x = $ 19.10
Or, price of three meals before tax = $ 19.10
so, The price of each meal = [tex]\frac{19.10}{3}[/tex] = $ 6.367
Hence The price of each meal cost before tax is $ 6.367 answer
1. In a survey sample of 83 respondents, about 30.1 percent of the sample work less than 40 hours per week. Calculate a 68 percent confidence interval for the proportion of persons who work less than 40 hours per week.
Answer:
A 68 percent confidence interval for the proportion of persons who work less than 40 hours per week is (0.251, 0.351), or equivalently (25.1%, 35.1%)
Step-by-step explanation:
We have a large sample size of n = 83 respondents. Let p be the true proportion of persons who work less than 40 hours per week. A point estimate of p is [tex]\hat{p} = 0.301[/tex] because about 30.1 percent of the sample work less than 40 hours per week. We can estimate the standard deviation of [tex]\hat{p}[/tex] as [tex]\sqrt{\hat{p}(1-\hat{p})/n}=\sqrt{0.301(1-0.301)/83} = 0.0503[/tex]. A [tex]100(1-\alpha)%[/tex] confidence interval is given by [tex]\hat{p}\pm z_{\alpha/2}\sqrt{\hat{p}(1-\hat{p})/n[/tex], then, a 68% confidence interval is [tex]0.301\pm z_{0.32/2}0.0503[/tex], i.e., [tex]0.301\pm (0.9944)(0.0503)[/tex], i.e., (0.251, 0.351). [tex]z_{0.16} = 0.9944[/tex] is the value that satisfies that there is an area of 0.16 above this and under the standard normal curve.
Final answer:
To calculate the 68 percent confidence interval for the proportion of persons working less than 40 hours per week from a sample of 83 respondents with a sample proportion of 30.1 percent, we use the formula for the confidence interval for a proportion. The resulting interval is approximately 25.06% to 35.14%.
Explanation:
To calculate a 68 percent confidence interval for the proportion of persons who work less than 40 hours per week from a sample of 83 respondents, where 30.1 percent work less than 40 hours, we use the formula for a confidence interval for a proportion:
In this formula:
p is the sample proportion (0.301 in this case).z* is the z-value corresponding to the desired confidence level (for 68 percent confidence, use the z-value corresponding to one standard deviation from the mean in a standard normal distribution, which is approximately 1).n is the sample size (83).Plugging the values into the formula we get:
0.301±1*sqrt((0.301(0.699)/83))
Calculating the square root part, we have:
0.301±1*sqrt((0.301*0.699)/83)
= 0.301±1*sqrt(0.210699/83)
= 0.301±1*sqrt(0.002539)
= 0.301±1*0.05039
= 0.301±0.05039
The confidence interval is thus:
0.301-0.05039 to 0.301+0.05039
= 0.25061 to 0.35139
Hence, with a 68 percent confidence level, we can say that the true proportion of the population that works less than 40 hours per week is estimated to be between 25.06% and 35.14%.
An Izod impact test was performed on 20 specimens of PVC pipe. The sample mean is and the sample standard deviation is s = 0.22. Find a 99% lower confidence bound on the true Izod impact strength. Assume the data are normally distributed.
For a 99% lower confidence bound, we use the Z-score of -2.33 with the formula for a confidence interval. The lower bound will be: 'Sample Mean - (-2.33) * (Sample Standard Deviation/√Sample Size).'
Explanation:The solution to this problem involves using concepts of statistics, primarily regarding normal distribution and confidence intervals. Given that we're finding a 99% lower confidence bound, we're interested only in the lower range of the spectrum, not the upper.
We need to look at the Z-score associated with a 99% confidence interval in a standard normal distribution. The Z-score for 99% is approximately 2.33 (meaning it cuts off the lowest 0.5% and highest 0.5% of the curve). However, since we're only interested in the lower bound, we will be using a Z-score of -2.33.
The formula for a confidence interval is: µ = X ± Z(s/√n), where µ is the population mean, X is the sample mean, Z is the Z-score, s is the sample standard deviation, and n is the size of the sample. In our question, X is unspecified, s = 0.22, and n = 20. So, assuming 'Xbar' is your sample mean, your lower bound would be: Xbar - (-2.33) * (0.22/√20)
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To find a 99% lower confidence bound on the true Izod impact strength, use the formula: Lower Confidence Bound = Sample Mean - (Critical Value × (Sample Standard Deviation / √n)). Substitute the given values into the formula and perform the necessary calculations.
Explanation:To find a 99% lower confidence bound on the true Izod impact strength, we can use the formula:
Lower Confidence Bound = Sample Mean - (Critical Value × (Sample Standard Deviation / √n))
Given that the sample mean is ______ and the sample standard deviation is s = 0.22, we need to calculate the critical value for a 99% confidence level. The critical value for a 99% confidence level is approximately 2.576. Plug in the values into the formula, substituting n = 20:
Lower Confidence Bound = ______ - (2.576 × (0.22 / √20))
Perform the necessary calculations to find the lower confidence bound.
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If A, B, and C are the measures of the angles of any triangle and if a, b, and c are the lengths of the sides opposite the corresponding angles, then which of the following expressions does not represent the area of the triangle?
A. (1/2)bcsinA
B. (1/2)acsinB
C. (1/2)acsinA
D. (1/2)absinC
Answer:
C. (1/2)acsinA
Step-by-step explanation:
Given is that, A, B, and C are the measures of the angles of a triangle and a, b, and c are the lengths of the sides opposite the corresponding angles.
So, the expression that does not represent the area of the triangle is :
C. (1/2)acsinA
The doubling time of a population of flies is 4 hours . By what factor does the population increase in 48 hours?
Answer:
4096
Step-by-step explanation:
The population doubles 12 times in that period, so is multiplied by 2^12 = 4096.
According to a candy company, packages of a certain candy contain 24% orange candies. Find the approximate probability that the random sample of 200 candies will contain 26% or more orange candies.
Final answer:
To find the approximate probability that the random sample of 200 candies will contain 26% or more orange candies, we can use the binomial distribution.
Explanation:
To find the approximate probability that the random sample of 200 candies will contain 26% or more orange candies, we need to use the binomial distribution.
The probability of selecting an orange candy is 24%. Let's define success as selecting an orange candy.
Using the binomial distribution formula, we can calculate the probability:
P(X ≥ k) = 1 - P(X < k)
where X is the random variable representing the number of orange candies in the sample, and k is the number of orange candies we want to have (26% of 200 is 52).
First, let's calculate P(X < 52). We can use a binomial probability table, or a binomial calculator to find this value. Once we have P(X < 52), we can find P(X ≥ 52) by subtracting it from 1.
A normal distribution is observed from the times to complete an obstacle course. The mean is 69 seconds and the standard deviation is 6 seconds. Using the Empirical Rule, what is the probability that a randomly selected finishing time is greater than 87 seconds?
Answer:
P ( z > 87 ) < 0,0015
P ( z > 87 ) < 0,15 %
Step-by-step explanation:
Applying the simple rule that:
μ ± 3σ , means that between
μ - 9 = 60 and
μ + 9 = 78
We will find 99,7 of the values
And given that z(s) = 87 > 78 (the upper limit of the above mention interval ) we must conclude that the probability of find a value greater than 87 is smaller than 0.0015 ( 0r 0,15 %)
To determine the probability of a finish time greater than 87 seconds, we apply the Empirical Rule and find it equates to 3 standard deviations above mean, resulting in a probability of 0.15%.
Explanation:The question revolves around the use of the Empirical Rule to determine the probability in a normal distribution. The mean time to complete an obstacle course is given as 69 seconds with a standard deviation of 6 seconds. According to the Empirical Rule:
To find out the probability of a finishing time being greater than 87 seconds, we first determine how many standard deviations above the mean this is:
(87 - 69) / 6 = 3
This indicates that 87 seconds is 3 standard deviations above the mean. Using the Empirical Rule, if 99.7% of the data falls within three standard deviations, this would leave 0.3% (or 0.003 in decimal form) of the data outside, which would be the tails of the distribution (both ends combined). Since we are looking for the area above 87 seconds, we only consider one tail, hence, we divide the 0.3% equally for each tail to get 0.15% (or 0.0015 in decimal form) for the probability that a randomly selected finish time is greater than 87 seconds.
Charles and Lisa were having a apple eating contest. They ate eighteen apples between the two of them. Lisa ate two more apples than Charles. How many apples did Lisa eat?
A.
6
B.
10
C.
12
D.
15
Answer:10
Step-by-step explanation:
ok they are 18 apples
and they ate them
one ate two more thatn the other one
so we can say that one of them is x(charles)
and the one who ate two more is x+2(lisa)
so we get this
x+x+2=18
2x+2=18
2x+18-2
2x=16
x=8
so charles ate 8
and lisa 10(8+2)
Answer:
1000% its 10
Step-by-step explanation:
Express each statement using an inequality involving absolute value: The height of the plant must be within 2 inches of the standard 13-inch show size
Answer:
|h-13| ≤ 2
Step-by-step explanation:
The difference between the height of the plant (h) and show size (13 in) can be written as ...
h - 13
This value is allowed to be positive or negative, but its absolute value must not exceed 2 inches. Thus, the desired inequality is ...
|h -13| ≤ 2
To express the statement using an Inequality involving absolute value, use |height - 13| ≤ 2. This means the difference between the height of the plant and 13 inches must be less than or equal to 2 inches.
To express the statement using an inequality involving absolute value, we can use the inequality |height - 13| ≤ 2. This means that the difference between the height of the plant and 13 inches must be less than or equal to 2 inches.
For example, if the height of the plant is 12 inches, then |12 - 13| = |-1| = 1, which is less than 2, so it satisfies the inequality. However, if the height of the plant is 16 inches, then |16 - 13| = |3| = 3, which is not less than 2, so it does not satisfy the inequality.
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An insurance agent receives 16% commission on every premium paid. how much commission is paid on a policy with a premium of $497.69? $31.10 $79.63 $311.06 $796.30
Answer:
$79.63
Step-by-step explanation:
You can figure this by estimating. 16% is a little less than 20%, which is 1/5. $497.69 is almost $500. So, 1/5 of that is almost $100, and a little less than that is about $80. The closest answer choice is $79.63.
__
If you want to figure it exactly, you can do the multiplication ...
16% of $497.69 = 0.16 × $497.69 = $79.6304 ≈ $79.63