A square has all 4 sides equal, so divide the amount of fence available by 4 to get the length of one side of the square
600/4 = 150
Now since the creek is being used for one side, add one side of the square to the other side to get a rectagle 150 by 300 feet.
Area = 150 x 300 = 45,000 square feet.
The maximum possible area of the pasture is;
A_max = 45000 ft²
We are given;Available fencing; Perimeter = 600 feet
Number of sides to fence; 3 sides of rectangle
Since we are dealing with rectangle, let L be the length and W be the width.
Perimeter of rectangle; P = 2L + 2W
But we are told one of the edges is the creek.
Thus, New perimeter = L + 2W
thus, we have; L + 2W = 600
L = 600 - 2W
Formula for Area of a rectangle is; A = LW
Let's put 600 - 2W for L in the area equation to get;
A = (600 - 2W)W
A = 600W - 2W²
We can maximize this area by finding the value of W when dA/dW = 0
Thus;
dA/dW = 600 - 4W
At dA/dW = 0, we have;
600 - 4W = 0
4W = 600
W = 600/4
W = 150 ft
Let's put 150 for W in L = 600 - 2W
L = 600 - 2(150)
L = 600 - 300
L = 300 ft
Therefore, Maximum possible area of pasture = 300 × 150 = 45000 ft²
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Can anyone help me with these two problems? I found the distance to one. Please, assignment is due tonight. Thank you.
Answer:
32 mi @ N77°E
Step-by-step explanation:
The route of travel forms legs of a right triangle, so the final distance (d) from harbor can be found using the Pythagorean theorem:
d² = 28² +16² = 784 +256 = 1040
d = √1040 ≈ 32.249 . . . miles
___
The angle (α) added to the original N47°E bearing can be found using the fact that opposite and adjacent sides are given. So, they can tell you the tangent of the angle:
tan(α) = 16/28
α = arctan(16/28) ≈ 29.74°
Then the bearing to the final position is ...
47° +29.74° = 76.74° . . . . . east of north
Rounded to whole numbers, the final position from harbor is ...
32 miles @ N 77° E
Mr. Johnson has his newspaper delivered every day. He determines the probability the newspaper is on his front porch is 0.64. The rest of the time he finds the newspaper somewhere on his lawn. What is the probability that Mr. Johnson finds his newspaper on his lawn?
Answer:
Step-by-step explanation:
Probability of getting the newspaper on his front porch = P(A) = 0.64
Probability of finding the newspaper on his lawn = P(B) =? (We don't know, this is what we have to find.)
Total probability = 1
Formula, P(A) + P(B) = 1
Let's put all values in this formula,
0.64 + P(B) = 1
P(B) = 1 - 0.64
P(B) = 0.36
Thus, the probablity that Mr. Johnson finds his newspaper on his lawn is 0.36 or 36%.
How do you solve this? 5/14 divided by 4/7??
Answer:
The answer to your question is: 5/8
Step-by-step explanation:
data
5/14 divided by 4/7
Process
[tex]\frac{5}{14\\}[/tex]
[tex]\frac{4}{7}[/tex]
multiply 5 and 7 and the result is the numerator
multiply 14 and 4 and the result is the denominator
(5 x 7) / ( 14 x 4)
35 / 56
5 / 8 simplify both numerator and denominator
Many states have programs for assessing the skills of students in various grades. The Indiana Statewide Testing for Educational Progress (ISTEP) is one such program. In a recent year, 76,531 tenth-grade Indiana students took the English/language arts exam. The mean score was 572 and the standard deviation was 51. Use the fact that the ISTEP scores are approximately Normal, N(572, 51). Find the proportion of students who have scores between 500 and 650.
Answer:
P ( 500<X<650 ) = 0.8577
Step-by-step explanation:
Since μ=572 and σ=51 we have:
P ( 500<X<650 ) = P ( 500−572< X−μ<650−572 )
[tex]\RightarrowP ( \frac{500-572}{51} < \frac{x-\mu}{\sigma} < \frac{650-572}{51})[/tex]
⇒ P ( 500<X<650 ) = P ( −1.41<Z<1.53 )
Now, Using the standard normal table to conclude that:
P ( −1.41< Z <1.53 ) = 0.8577
Final answer:
Approximately 85.77% of Indiana tenth-grade students scored between 500 and 650 on the English/language arts ISTEP exam, as calculated by converting the given scores to z-scores and finding the proportion within the standard normal distribution.
Explanation:
To find the proportion of Indiana tenth-grade students who scored between 500 and 650 on the English/language arts exam, given that the ISTEP scores are approximately normal with a mean (μ) of 572 and a standard deviation (σ) of 51, we can use the standard normal distribution.
First, we'll convert the scores to z-scores using the formula z = (X - μ) / σ where X is the score.
For X=500, the z-score would be (500 - 572) / 51 ≈ -1.41.For X=650, the z-score would be (650 - 572) / 51 ≈ 1.53.Then we'll look up these z-scores in a standard normal distribution table or use a calculator with normal distribution functions to find the proportion of students whose scores fall between these two z-scores. The table or calculator will provide us with the areas under the curve to the left of each z-score.
Let's assume the area to the left of z=-1.41 is approximately 0.0793 (7.93%) and to the left of z=1.53 is approximately 0.937 (93.7%). To find the proportion between 500 and 650, we'll subtract the smaller area from the larger one:
Proportion = Area to the left of z=1.53 - Area to the left of z=-1.41 = 0.937 - 0.0793 = 0.8577 or 85.77%
Therefore, approximately 85.77% of the students scored between 500 and 650 on the ISTEP English/language arts exam.
Which is the graph of f(x) = Square root of x?
On a coordinate plane, a parabola opens up with a vertex at (0, 0).
On a coordinate plane, an absolute value graph starts at (0, 0) and goes up through (2, 4).
On a coordinate plane, a parabola opens to the right with a vertex at (0, 0).
On a coordinate plane, an absolute value graph starts at (0, 0) and goes up through (4, 2).
Answer:
Third option given
Step-by-step explanation:
On a coordinate plane, a parabola opens to the right with a vertex at (0, 0).
use the following x values to plot the associated y values on the x-y plane:
If x = 0, then [tex]y=\sqrt{0} =0[/tex]
If x = 1, then [tex]y=\sqrt{1} =1[/tex]
If x = 4, then [tex]y=\sqrt{4} =2[/tex]
If x = 9, then [tex]y=\sqrt{9} =3[/tex]
Join the points and you will find a branch of a parabola opening to the right.
Answer:
The Answer is D
Step-by-step explanation:
on e2020 its the last graph
If the number of $8 child tickets is 17 less than 3 times the number of $12 adult tickets and the theater took in $584, how many of each ticket were sold?
Answer:
The number of child tickets sold was 43 and the number of adult tickets sold was 20
Step-by-step explanation:
Let
x ----> the number of child tickets sold
y ----> the number of adult tickets sold
we know that
[tex]x=3y-17[/tex] -----> equation A
[tex]8x+12y=584[/tex] ----> equation B
Solve the system by graphing
Remember that the solution is the intersection point both graphs
The solution is the point (43,20)
see the attached figure
therefore
The number of child tickets sold was 43 and the number of adult tickets sold was 20
Answer:
Number of child tickets sold = 43
Number of adult tickets sold = 20
Step-by-step explanation:
Let the number of child ticket be c and number of adult ticket be a,
Given that the number of $8 child tickets is 17 less than 3 times the number of $12 adult tickets,
c = 3a - 17
3a - c = 17 ----------------------eqn 1
The theater received $584
That is
8 c + 12 a = 584 ----------------------eqn 2
eqn 2 /4
2 c + 3 a = 146 ----------------------eqn 3
eqn 3 - eqn 1 gives
2 c + 3 a - (3a - c) = 146 - 17
3 c = 129
c = 43
Substituting in eqn 1
3 x a - 43 = 17
3a = 60
a = 20
Number of child tickets sold = 43
Number of adult tickets sold = 20
PLEASE I BEG U NOW!!
£130 is divided between Henry, Gavin & Jim so that Henry gets twice as much as Gavin, and Gavin gets three times as much as Jim. How much does Henry get?
Answer:
The answer to your question is: Henry gets £ 78
Step-by-step explanation:
Data
Henry = H
Gavin = G
Jim = J
Henry + Gavin + Jim = £ 130
H = 2G
G = 3J
Process
Write an equation in terms of G
H + G + J = 130
2G + G + G/3 = 130
Solve it for G
3( 2G + G + G/3 = 130)
6G + 3G + G = 390
10G = 390
G = 390 / 10
G = £39
H = 2G = 2(39)
H = £78
J = G/3
J = 39/3
J = £13
Final answer:
By setting up and solving algebraic equations, we find that Henry receives £78, which is twice the amount that Gavin receives and six times the amount that Jim receives from the total sum of £130.
Explanation:
Let's solve the problem by denoting Jim's share of the money as x. According to the problem, Gavin gets triple the amount of money that Jim gets, which means Gavin gets 3x. Henry gets twice as much as Gavin, so he gets 2(3x) or 6x. The total amount of money is £130 and the sum of their shares is x + 3x + 6x = 10x. Setting up the equation:
10x = £130,
we find that x, Jim's share, is £130/10 = £13. Gavin's share is 3 times Jim's: 3 × £13 = £39, and Henry's share is 6 times Jim's: 6 × £13 = £78. So Henry gets £78.
If we have the curve y = sqrt(x), find the y value and the slope of the curve when x = 36. y = 6 Correct: Your answer is correct. slope = 1/12 Correct: Your answer is correct. Hence, find the equation of the tangent line to the curve at x = 36, writing your answer in the form y = mx + c. What are the values of m and c?
m =
c =
Answer:
Equation of tangent of curve at x = 36:
1)[tex]y = \frac{x}{12} + 3[/tex]
2[tex]y = \frac{-x}{12} - 3[/tex]
Step-by-step explanation:
We are given the following information:
[tex]y = \sqrt{x}[/tex]
Value of curve when x = 36:
[tex]y = \sqrt{36} = \pm 6[/tex]
Thus, [tex]y = \pm6[/tex], when x = 6.
Slope of curve, m =
[tex]\frac{dy}{dx} =\frac{d(\sqrt{x})}{dx}=\frac{1}{2\sqrt{x}}[/tex]
At x = 36,
slope of curve =
[tex]\frac{1}{2\times \sqrt{36}}\\\\m=\frac{1}{12},\frac{-1}{12}[/tex]
Equation of tangent of curve at x = 36:
[tex](y-y_1) = m(x-x_1)[/tex]
[tex]= (y-(\pm 6)) = (\pm\frac{1}{12} )(x - 36)[/tex]
Thus, equation of tangents are:
1)
[tex](y-6) = \frac{1}{12}(x-36)\\12(y-6) = x-36\\y = \frac{x}{12} + 3[/tex]
Comparing to [tex]y = mx + c[/tex], we get [tex]m = \frac{1}{12}[/tex] and [tex]c =3[/tex]
2)
[tex](y+6) = \frac{-1}{12}(x-36)\\12(y+6) = -x+36\\y = \frac{-x}{12} - 3[/tex]
Comparing to [tex]y = mx + c[/tex], we get [tex]m = \frac{-1}{12}[/tex] and [tex]c =-3[/tex]
Final answer:
The equation of the tangent line at x = 36 for the curve y = sqrt(x) is y = 1/12x + 3, found by using the slope of 1/12 and solving for the y-intercept c, which is 3.
m = 1/12
c = 3
Explanation:
To find the equation of the tangent line at x = 36 for the curve y = sqrt(x), we first identified that y = 6 when x = 36 and computed the slope of the curve, which is 1/12 at x = 36.
Using the slope-intercept form of a line, y = mx + c, where m is the slope, and c is the y-intercept, we can substitute m = 1/12 and the point (36, 6) to solve for c.
Substituting into the slope-intercept equation gives us 6 = (1/12)\(36) + c.
Solving for c, we find that the y-intercept c = 3.
Thus, the equation of the tangent line is y = 1/12x + 3.
Jim Murray and Phyllis Lowe received a total of $52,000 from a deceased relative's estate. They decided to put $10,400 in a trust for their nephew and divide the remainder. Phyllis received 45 of the remainder; Jim received 15. How much did Jim and Phyllis receive?
Answer:
Phyllis will get $33280.
Jim will get $8320.
Step-by-step explanation:
Jim Murray and Phyllis Lowe received a total of $52,000 from a deceased relative's estate. They decided to put $10,400 in a trust for their nephew and divide the remainder.
The remainder = [tex]52000-10400=41600[/tex] dollars
Phyllis received 4/5 of the remainder; = [tex]0.8 \times41600=33280[/tex] dollars
Jim received 1/5 ; =[tex]0.2\times41600=8320[/tex] dollars
So, Phyllis will get $33280.
Jim will get $8320.
Phyllis received $18,720, which was 45% of the remaining estate, while Jim received $6,240, which was 15% of the remaining estate.
Jim Murray and Phyllis Lowe received a total of $52,000 from a deceased relative's estate. They first put $10,400 in a trust for their nephew. This leaves $52,000 - $10,400 = $41,600 remaining to divide between Jim and Phyllis. Phyllis received 45% of the remainder and Jim received 15% of the remainder.
To find out how much Phyllis received, calculate 45% of $41,600, which is $41,600 x 0.45 = $18,720. To find out how much Jim received, calculate 15% of $41,600, which is $41,600 x 0.15 = $6,240. Therefore, Phyllis received $18,720 and Jim received $6,240 from the remaining estate after the trust was funded.
Today a typical family of four spends $897.20/ month for food. If inflation occurs at the rate of 3%/ year over the next 6 years, how much should the typical family of four expect to spend for food 6 years from now?
Answer:
1071.30 per month
Step-by-step explanation:
The multiplier each year is 1 + .03 = 1.03. After 6 years, the cost has been multiplied by that factor 6 times, so has been multiplied by 1.03⁶ ≈ 1.194052.
$897.20 × 1.194052 ≈ $1071.30
The typical family of four should expect to spend $1,056.27 per month for food six years from now.
1. Calculate the inflation factor using the formula: [tex]\( (1 + \text{inflation rate})^{\text{number of years}} \).[/tex]
- Inflation rate = 3% or 0.03
- Number of years = 6
- Inflation factor = [tex]\( (1 + 0.03)^6 = 1.191016 \)[/tex]
2. Multiply the current monthly food expenditure by the inflation factor to find the expected expenditure six years from now.
- Current expenditure = $897.20/month
- Expected expenditure = $897.20 × 1.191016 = $1,056.27/month
Therefore, the typical family of four should expect to spend $1,056.27 per month for food six years from now if inflation occurs at a rate of 3% per year.
If DE = 5x EF = 3x and DF = 32 what is the length of DE?
Answer:
15
Step-by-step explanation:
Answer:
20
Step-by-step explanation:
Combine like terms: 5x + 3x = 8x
Divide both sides by 8: 8x/4; 32/8
Solve for x: x = 4
Plug it in: DE = 5(4) = 20
Check: 5(4) + 3(4) = 32; 20 + 12 = 32
1.Solve for t.
3t + 35=6
Enter your answer in the box.
t =
Answer:
3t + 35 = 6
subtract 35 from both the sides of the equation
3t + 35 - 35 = 6-35
3t = -29
t = - 29/3
Step-by-step explanation:
Answer:
-29/3
Step-by-step explanation:
3t+35=6
3t=6-35
3t=-29
t=-29/3
Sarah has 2/3 gallon of blue paint and 7/12 gallon of red paint. If she has a total of 2 1/8 gallons of paint, how many gallons are neither red nor blue?
Suppose that 0 < c < π/2. For what value of c is the area of the region enclosed by the curves y = cos(x), y = cos(x − c), and x = 0 equal to the area of the region enclosed by the curves y = cos(x − c), x = π, and y = 0? (Note: These areas are in the interval 0 ≤ x ≤ π.)
Answer:
[tex]c = \cfrac{\pi}{3}[/tex]
Step-by-step explanation:
We can see in the first figure that area enclosed between the curves is a triangle like shape that is between the x= 0 axis and the two cosine curves, whose top vertex is the point (0,1) . We can calculate its area by integration, but first we must find the intersection between cos(x) and cos (x-c)
We can check easily that the intersection between these curves will always be [tex]x= \cfrac{c}{2}[/tex] because cos is an even function.
[tex]\cos\Big(\cfrac{c}{2}-c\, \Big) = \cos\Big(-\cfrac{c}{2}\, \Big) = \cos\Big( \cfrac{c}{2}\, \Big)[/tex]
Where the last step is justified because cosine is an even function.
now, to find the area we will integrate the difference between these functions as follows:
[tex]A= \int\limits^{\frac{c}{2}}_0 {\cos(x) - \cos(x-c)} \, dx \\\\A= \Big[ \sin(x) - sin (x-c) \, \Big]^{c/2 }_0\\ \\A= \sin(c/2) - \sin (-c/2) - \big( sin (0) - sin (-c) \big) \\\\A= 2 \sin (c/2) - \sin (c)[/tex]
Where we have used the property that sine is an odd function.
Now we must find the area for the other region, which is shown on the bottom right in the second figure.
Now, we have to find for which value of x will the cos(x-c) intersect the y=0 axis. We can see tho that as the function cos(x-c) is just the cosine function displaced to the right by c units, its root will shift aswell. Therefore we see that when [tex]x= \cfrac{\pi}{2} + c[/tex] we will have that: [tex]\cos(x-c) = 0[/tex]
Now, to find the second area, (let's call it B) we integrate the difference between the top and bottom curves, in this case y= 0 and y= cos(x-c) up until [tex]x= \pi[/tex]
[tex]B=\int\limits^{\pi}_{\pi/2+c} {0-\cos(x-c)} \, dx \\\\B= \Big[ - \sin(x-c)\Big]^\pi_{\pi/2+c}\\\\B= \sin(\pi/2)-\sin(\pi-c)[/tex]
And finally we set A = B:
[tex]2 \sin (c/2) - sin (c) = 1 - \sin(c)\\\\2 \sin (c/2)= 1\\\\ \sin (c/2) = \cfrac{1}{2} \implies c/2 = \pi /6 \\\\\implies c = \cfrac{\pi}{3}[/tex]
This problem involves finding the value of 'c' such that two areas under different curves are equal. This requires setting up and solving two integral equations, which involves knowledge of trigonometric functions and calculus. This is a challenging problem.
Explanation:The question asks for the value of c when the first region enclosed by the curves y = cos(x), y = cos(x - c), and x = 0 has equal area to the second region enclosed by the curves y = cos(x - c), x = π, and y = 0. To solve this, you need knowledge about trigonometric functions and properties. This is less about probability and more about areas under curves and integral calculus.
Generally, if you're asked to find areas under curves, the area between the x-axis (in this case, y=0) and the curve from x=a to x=b is given by the definite integral ∫ from a to b of f(x) dx, where f(x) is the equation of the curve. If you have not studied calculus, you might struggle with this problem because it essentially asks you to set up and solve two different integral equations and then to equate those two areas in order to solve for c.
Please seek additional help in setting up and solving these integrals if you need it. This is a challenging problem!
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An equation of the secant line containing (4, h(4)) and (6, h(6)) is
Answer:
y= (h(6)-h(4))/2x +C
Where C=-2*h(6)+3*h(4)
Step-by-step explanation:
The secant line is the line that meets a function (in this case h(x) ) in two points, so we have to apply the ecuation of a straigt line that meets two points:
y-y1 = (y2-y1)/(x2-x1) * (x-x1)
In this case X1=4 , x2=6, y1 = h(4) and y2= h(6)
So
y-h(4)= 1/2 (h(6)-h(4)) * (x-4)
y-h(4)= 1/2 (h(6)-h(4)) x- 2 *(h(6)-h(4))
y-h(4)= 1/2 (h(6)-h(4)) x- 2 (h(6) + 2h(4))
y= 1/2 (h(6)-h(4)) x- 2 h(6) + 2h(4) + h(4)
y= 1/2 (h(6)-h(4)) x- 2 h(6) + 3h(4)
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what is the area of the composite figure 262 264 266 268
Answer:
264cm2
Step-by-step explanation:
the length of line in green = 5 cm
the length of line in yellow = 3 cm
the length of line in red = 6 cm
the length of line in white = 16 cm
the total area = 100 + 12 + 12 + 20 + 24 + 96 = 264 cm2
The area of the composite figure is 262 cm².
What is a square?A square is a two-dimensional figure that has four sides and all four sides are equal.
The area of a square is side².
We have,
The composite figure has:
- 3 squares of 10 cm, 4 cm, and 6 cm.
- 1 rectangle of 3cm x 10 cm
- 1 trapezium of two paralel sides (10 cm and 6 cm) and height of 7 cm.
- 1 rectangle with 4cm x 6cm.
Now,
Area of the squares.
= 10² + 4² + 6²
= 100 + 16 + 36
= 100 + 52
= 152 cm²
Area of the rectangle.
= 3 x 10
= 30 cm²
Area of trapezium.
= 1/2 x (10 + 6) x 7
= 1/2 x 16 x 7
= 56 cm²
Area of the rectangle.
= 4 x 6
= 24 cm²
Now,
Area of the composite figure.
= 152 + 30 + 56 + 24
= 262 cm²
Thus,
262 cm² is the area.
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In a sample of 1,600 registered voters, 912, or 57%, approve of the way the President is doing his job.
A political pollster states: "Fifty-seven percent of all voters approve of the President." This statement is an example of
a) A Sample
b) Descriptive Statistics
c) Statistical Inference
d) A Population
Answer:
This statement is an example of - statistical inference
Step-by-step explanation:.
A political pollster states: "Fifty-seven percent of all voters approve of the President."
This statement is an example of - statistical inference
Statistical inference is defined as the process of drawing conclusions about populations by using data analysis.
There are two types of statistical inference - statistical estimation and statistical hypothesis testing.
Consider the three functions below.
f(x) = Negative StartFraction 6 Over 11 EndFraction (eleven-halves) Superscript x g(x) = StartFraction 6 Over 11 EndFraction (eleven-halves) Superscript negative x h(x) = Negative StartFraction 6 Over 11 EndFraction (eleven-halves) Superscript negative x
Which statement is true?
The range of h(x) is y > 0.
The domain of g(x) is y > 0.
The ranges of f(x) and h(x) are different from the range of g(x).
The domains of f(x) and g(x) are different from the domain of h(x).
Answer:
The range of g(x) is y > 0
The ranges of f(x) and h(x) are different from the range of g(x)
Step-by-step explanation:
we have
[tex]f(x)=-(\frac{6}{11})^{x}[/tex]
[tex]g(x)=(\frac{6}{11})^{-x}[/tex]
[tex]h(x)=-(\frac{6}{11})^{-x}[/tex]
Using a graphing tool
see the attached figure
Verify each statement
case A) The range of h(x) is y > 0.
The statement is false
The range of h(x) < 0
case B) The range of g(x) is y > 0. (Note the statement is The range of g(x) is y > 0 instead of The domain of g(x) is y > 0)
The statement is true (see the attached figure)
case C) The ranges of f(x) and h(x) are different from the range of g(x)
The statement is true (see the attached figure)
Because
The ranges of f(x) and h(x) are y < 0
and
The range of g(x) is y > 0
case D) The domains of f(x) and g(x) are different from the domain of h(x)
The statement is false
The domain of the three functions is the same
Answer:
OPTION C.The ranges of f(x) and h(x) are different from the range of g(x).
Step-by-step explanation:
A map of a rectangular park has a length of 4 inches and a width olo and a width of 6 inches. It uses a scale of 1 inch for every 30 miles. a. What is the actual area of the park? Show how you know.
Answer:
The actual area of the park is 21,600 miles square.
Step-by-step explanation:
Step 1: Calculate the area of the park
Area of rectangle = Length x width
Length = 4 inches
Width = 6 inches
Scale factor is given as, 1 inch = 30 miles.
So, converting the length and width in miles by using the scale factor, we get
Length = 4 * 30 = 120 miles
Width = 6 * 30 = 180 miles
The formula of finding the area of an rectangle is:
Area = length x width
By putting the values in equation, we get
Area = 120 x 180
Area = 21,600 miles square
Therefore, the actual area of the park is 21,600 miles square.
solve for m.
4+|7-m|=5
Answer:
m= 6 or m= 8. I'm not sure if its multiple choice, but that's what I got.
Answer:
m=6,8
Step-by-step explanation:
Colton and Gage were building a castle together. Colton was 5 times as fast at building the castle than Gage. If it takes 120 blocks to build the castle, how many blocks did Colton build on the castle?
Answer:
100 blocks.
Step-by-step explanation:
Let x represent number of blocks built by Gage.
We have been given that Colton was 5 times as fast at building the castle than Gage, so number of blocks built by Colton would be [tex]5x[/tex].
We are also told that it takes 120 blocks to build the castle. Since Colton and Gage were building a castle together, so number of blocks built by both will be equal to 120.
[tex]x+5x=120[/tex]
[tex]6x=120[/tex]
[tex]\frac{6x}{6}=\frac{120}{6}[/tex]
[tex]x=20[/tex]
Number of blocks built by Colton: [tex]5x=5\cdot 20=100[/tex].
Therefore, Colton had build 100 blocks.
a linear system in three variables has no solution. Your friend concludes that it is not possible for two of the three equations to have any points in common. is your friend correct? Explain your reasoning.
Answer:
He is correct.
Step-by-step explanation:
Because a linear system in three variables means that they have solutions if the all three have points in common in R3 space. But, in the case that they don't have any solution mean the opposite, they don't have points in common in R3 space.
In a linear system, variables must be related through common points. So, graphically, you should draw intersecting geometrical places in order to show the intersections, wich are the common results or points that are the solutions of the linear system.
A linear system of equation in three variables has no solution is the correct statement.
What is linear equation?" Linear equation is defined as the equation whose variables with highest degree one."
According to the question,
Given statement,
A linear system of equation in three variables has no solution.
Verification:
Consider a example of linear system of equation with three variables
[tex]2x-4y +z=3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)[/tex]
[tex]8x-2y+ 4z=7 \ \ \ \ \ \ \ \ \ \ \ \ \ (2)[/tex]
[tex]-4x+y -2z=-14\ \ \ \ \ \ \ \ \ \ (3)[/tex]
Solve linear equation to get the solution
Multiply [tex](3)[/tex] by [tex]2[/tex] and add it to [tex](2)[/tex],
[tex]\ \ 8x-2y+ 4z=7\\\\-8x+2y-4z =-28[/tex]
we get,
[tex]0x+0y + 0z = -21[/tex] which is not possible and has no solution.
Hence, linear system of equation has no solution is the correct statement.
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An urn contains 10 balls: 7 red and 3 blue. A second urn contains 25 red balls and an unknown number of blue balls. A single ball is drawn at random from each urn. The probability that the balls are the same color is ________
Answer:
The probability that the balls are the same color is [tex]\frac{175-7x}{250}[/tex] where x is the number of blue balls in the second urn
Step-by-step explanation:
Let's call x the number of blue balls in the second urn. Since there are 25 ball in this urn, the number of red balls would be 25 - x.
Now, we're looking for the probability that the balls are the same color, this means, that both balls are either blue OR red. (since we're using "or", this gives us the clue that we will sum up both probabilities.
1) P(both balls are the same color) = P(both balls are blue) + P( both balls are red).
2) Now we will find what is the probability that both balls are blue (this will be a multiplication since we need that ball 1 is blue AND ball 2 is blue:
P(both balls are blue) = P(ball 1 is blue) x P(ball 2 is blue)
P(both balls are blue) = [tex](\frac{3}{10} )(\frac{x}{25} )[/tex] = [tex]\frac{3x}{250}[/tex]
3) To find what is the probability that both balls are red, the process is similar than when both are blue.
P(both balls are red) = P(ball 1 is red) x P(ball 2 is red)
P(both balls are red) = [tex](\frac{7}{10} )(\frac{25-x}{25} )[/tex]= [tex]\frac{175-7x}{250}[/tex]
4) Going back to 1) and substituting:
P(both balls are the same color) = [tex]\frac{3x}{250} +\frac{175-7x}{250} \\ \\ = \frac{3x-7x+175}{250} \\ \\=\frac{175-4x}{250}[/tex]
Final answer:
The probability that the balls are the same color is 17/40.
Explanation:
The probability that the balls are the same color is 17/40.
To calculate this, we need to find the probabilities of both balls being red and both balls being blue and then add them together.
Probability of both balls being red: (7/10) * (26/35) = 182/350
Probability of both balls being blue: (3/10) * (9/35) = 27/350
Finally, add the two probabilities: 182/350 + 27/350 = 209/350 = 17/40
In regional spelling bee, the 8 finalists consist of 3 boys and 5 girls. Find the number of sample point the sample space S for the number of possible orders at the conclusion of the contest for:
- All 8 finalist
- The fist 3 positions
Answer:
i) There are 40320 possible orders
ii) There are 336 possible orders for the first 3 positions.
Step-by-step explanation:
Given: The number of finalists = 8
The number of boys = 3
The number of girls = 5
To find the number of sample point the sample space S for the number of possible orders, we need to find factorial of 8!
The number of possible orders = 8!
= 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8
= 40320
ii) From all 8 finalist, we need to choose first 3 position. Here the order is important. So we use permutation.
nPr =[tex]\frac{n!}{(n - r)!}[/tex]
Here n = 8 and r = 3
Plug in n =8 and r = 3 in the above formula, we get
8P3 = [tex]\frac{8!}{(8 - 3)!}[/tex]
= [tex]\frac{8!}{5!} \\= \frac{1.2.3.4.5.6.7.8}{1.2.3.4.5}[/tex]
= 6.7.8
= 336
So there are 336 possible orders for the first 3 positions.
A salesperson obtained a systematic sample of size 25 from a list of 500 clients. To do so, he randomly selected a number from 1 to 20, obtaining the number 13. He included in the sample the 13th client on the list and every 20th client thereafter. List the numbers that correspond to the 25 clients selected.
A.. 25, 45.......500
B. 20, 33,.., 332
C. 13, 33, , 488
D. 13, 33, ..., 493
Answer: Option 'D' is correct.
Step-by-step explanation:
Since we have given that
First term = 13
Increase each time = 20
Number of clients = 25
So, it forms an arithmetic progression:
So, 25 th term would be
[tex]a_{25}=a+(n-1)d\\\\a_{25}=13+(25-1)\times 20\\\\a_{25}=13+24\times 20\\\\a_{25}=13+480\\\\a_{25}=493[/tex]
Hence, list would look like 13,33,............493.
Therefore, Option 'D' is correct.
Jon’s picture frame is 6 inches high by 5 inches wide. He wants to know what other size frames he can use if he enlarges the photo by using the same relative proportions for length and width.
How can Jon find the sizes of other frames?
Jon can plot the ratio of 5 to 6 on a graph (5, 6), then move up 6 and left 5 to find the next frame size, and then keep the same rate to find other points.
Jon can plot the ratio of 5 to 6 on a graph (5, 6), then move up 6 and right 5 to find the next frame size, and then keep the same rate to find other points.
Jon can plot the ratio of 6 to 5 on a graph (6, 5), then move right 6 and down 5 to find the next frame size, and then keep the same rate to find other points.
Jon can plot the ratio of 6 to 5 on a graph (6, 5), then move right 5 and up 6 to find the next frame size, and then keep the same rate to find other points.
Answer:
Jon can plot the ratio of 5 to 6 on a graph (5, 6), then move up 6 and right 5 to find the next frame size, and then keep the same rate to find other points.
Step-by-step explanation:
A line from the origin to the point (5, 6) has a slope of 6/5. To find other points on the same line, one would need to continue to rise 6 units for each run of 5 units to the right.
The only answer choice that appropriately matches (x, y) values to rise/run values is the one shown above.
_____
Comment on this answer
The slope of the line described in this answer is 6/5, which seems appropriate for a frame that is 6 inches high and 5 inches wide. However, I would call this a plot of the ratio 6 to 5, rather than the ratio 5 to 6.
Answer:
Jon can plot the ratio of 5 to 6 on a graph (5, 6), then move up 6 and right 5 to find the next frame size, and then keep the same rate to find other points.
i hope this helps!
Elvira and Aletheia live 3.1 miles apart on the same street. They are in a study group that meets at a coffee shop between their houses. It took Elvira 12 hour and Aletheia 23 hour to walk to the coffee shop. Aletheia's speed is 0.6 miles per hour slower than Elvira's speed. Find both women's walking speeds.
Answer:
Elvira's speed: 3.0 mphAletheia's speed: 2.4 mphStep-by-step explanation:
Let "e" and "a" represent the speeds of Elvira and Aletheia, respectively. Then the total distance they cover is ...
distance = speed × time
(1/2)e + (2/3)a = 3.1
And the relationship between their speeds is ...
e - a = 0.6
__
To solve this system, we can double the first equation and subtract the second to get ...
2(1/2e +2/3a) -(e -a) = 2(3.1) -(0.6)
7/3a = 5.6 . . . . . . . . . . simplify
a = (3/7)(5.6) = 2.4 . . . multiply by 3/7
e = a +0.6 = 3.0 . . . . . Elvira's speed is 0.6 mph more than Aletheia's
Elvira's walking speed is 3.0 miles per hour; Aletheia's is 2.4 miles per hour.
Genghis Khan organized his men into groups of 10 soldiers under a "leader of 10." Ten "leaders of 10" were under a "leader of 100." Ten "leaders of 100" were under a "leader of 1000." *(a) If Khan had an army of 10,000 soldiers at the lowest level, how many men in total were under him in his organization? (b) If Khan had an army of 5763 soldiers at the lowest level, how many men in total were under him in his organization? Assume that the groups of 10 should contain 10 if possible, but that one group at each level may need to contain fewer.
Genghis Khan's army organizational structure can be used to calculate the total number of men under him: For an army of 10,000 soldiers, the total is 11,110 men; for an army of 5,763 soldiers, it is 6,404 men.
To solve both parts of the student's question, we use the military organizational structure implemented by Genghis Khan that is based on the 'decimal' system. We start with the lowest level groups of 10 soldiers and move up in orders of magnitude (10, 100, 1,000, and 10,000).
Part (a): Army of 10,000 soldiers
Groups of 10: There are 1,000 groups of 10 in an army of 10,000 soldiers.
Leaders of 10: Each group of 10 has 1 leader, resulting in 1,000 leaders of 10.
Groups of 100: 1,000 leaders of 10 make up 100 groups of 100 (because 1,000/10 = 100).
Leaders of 100: There are 100 leaders of 100.
Groups of 1,000: 100 leaders of 100 make up 10 groups of 1,000 (because 100/10 = 10).
Leaders of 1,000: There are 10 leaders of 1,000.
Adding it all together: 10,000 soldiers + 1,000 leaders of 10 + 100 leaders of 100 + 10 leaders of 1,000 = 11,110 total men under the command of Khan for an army of 10,000.
Part (b): Army of 5,763 soldiers
Groups of 10: There are 576 full groups of 10 and 1 incomplete group (with 3 soldiers), resulting in 577 groups.
Leaders of 10: There are 577 leaders of 10.
Groups of 100: 577 leaders of 10 make up 57 full groups of 100 and 1 incomplete group (with 7 leaders of 10), resulting in 58 groups.
Leaders of 100: There are 58 leaders of 100.
Groups of 1,000: 58 leaders of 100 make up 5 full groups of 1,000 and 1 incomplete group (with 8 leaders of 100), resulting in 6 groups.
Leaders of 1,000: There are 6 leaders of 1,000.
Adding it all together: 5,763 soldiers + 577 leaders of 10 + 58 leaders of 100 + 6 leaders of 1,000 = 6,404 total men under the command of Khan for an army of 5,763 soldiers.
A committee organizing a marathon has 14 jugs of water and 20 jugs of sports drink. The committee would like to set up a number of refreshment stations along the marathon course, with the same combination of jugs of water and jugs of sports drink at each station, with no beverages left over. What is the greatest number of refreshment stations that can be set up?
Answer:
2 stations with 7 of jugs of water and 10 of sports drink in each station
Step-by-step explanation:
Lets start with the jugs of water, to have the same amount in each station you need have a number of station that divide for 14 the remainder or "left over" equal to 0, or in other words multiple of 14 in the interval 1 to 14
multiples of 14 : 1, 2, 7 and 14
so you can have
1 station with 14 jugs of water
2 stations with 7 jugs of water each
7 stations with 2 jugs of water each
14 station with 1 jugs of water each
Now we need to do the same for the 20 jugs of sports drink
multiple of 20 in the interval 1 to 20: 1,2,4,5,10 and 20
so you can have:
1 station with 20 jugs of sports drink
2 stations with 10 jugs of sports drink
4 stations with 5 jugs of sports drink
5 station with 4 jugs of sports drink
10 station with 2 jugs of sports drink
20 stations with 1 jugs of sports drink
So we have that both cases, water and sport drinks have coincidence of 1 or 2 stations, and the maximum of both is 2 stations with 7 jugs of water and 10 of sports drink each
Final answer:
To determine the greatest number of refreshment stations that can be set up, we find the greatest common divisor (GCD) of 14 jugs of water and 20 jugs of sports drink, which is 2. Therefore, the committee can set up 7 refreshment stations with 1 jug of water and 2 jugs of sports drink at each.
Explanation:
The committee has 14 jugs of water and 20 jugs of sports drink and wants to set up the greatest number of refreshment stations along the marathon course without any beverages left over. To find this solution, we need to determine the greatest common divisor (GCD) of the two quantities of jugs because each station must have the same combination of jugs of water and sports drink.
First, list the factors of each number:
Factors of 14: 1, 2, 7, 14
Factors of 20: 1, 2, 4, 5, 10, 20
The largest common factor is 2, which means the refreshment stations can have a combination of 1 jug of water and 2 jugs of sports drink. So, the committee can set up 7 refreshment stations (14 jugs of water / 2 per station = 7 stations; and 20 jugs of sports drink / 2 per station = 10 stations, but limited by the smaller number of jugs of water).
A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16 oz. serving size. The sample mean is 13.80 with a sample standard deviation of 1.57. Assume the underlying population is normally distributed. Find the 95% Confidence Interval for the true population mean for the amount of soda served.
a. (12.42, 14.18)
b. (12.32, 14.29)
c. (12.50, 14.10)
d. None of the above
e. Impossible to determine
Answer:
The answer is E. Impossible to determine
Step-by-step explanation:
Normally, you would find the Confidence interval of a normal sample by using
X(-+) Z* Sigma/n
Where x is the mean, sigma the standard deviation n the size of the sample and z the value determined by your confidence interval size of 95%
.However, this approximation of a confidence interval may only be used for a sample if the number of observations is at least 30 or above. When we have less observations than 30 we must use the standard deviation of the populations. But we only have a sample standard deviation so its not adequate or possible to determine CI the true mean of the population with such a small sample size.