Answer:
15 trees should be added to the existing orchard in order to maximize the total output of trees.
Step-by-step explanation:
Number of trees = 50
Number of apples each tree gives = 800
For each additional tree planted in the orchard, the output per tree drops by 10 apples. Let x be the number of trees added, then,
Number of apple is given by:
[tex]V(x) = (50+x)(800-10x) = -10x^2+300x + 40000 [/tex]
First, we differentiate V(x) with respect to x, to get,
[tex]\displaystyle\frac{d(V(x))}{dx} = \frac{d(-10x^2+300x + 40000)}{dx} = -20x + 300[/tex]
Equating the first derivative to zero, we get,
[tex]\displaystyle\frac{d(V(x))}{dx} = 0\\\\-20x + 300 = 0[/tex]
Solving, we get,
[tex]-20x + 300 = 0\\\\x=\displaystyle\frac{300}{20} = 15[/tex]
Again differentiation V(x), with respect to x, we get,
[tex]\displaystyle\frac{d^2(V(x))}{dx^2} = -20[/tex]
At x = 15
[tex]\displaystyle\frac{d^2(V(x))}{dx^2} < 0[/tex]
Thus, by double derivative test, the maxima occurs at x = 15 for V(x).
Thus, 15 trees should be added to the existing orchard in order to maximize the total output of trees.
Maximum output of apples =
[tex]V(15) = (50+15)(800-10(15)) = 42250[/tex]
Alexis raises 75.23 for charity.Sue raises 3 times as much as alexia.Manuel raises 85.89.How much money do the three raise for charity in all? Show work
Answer: the amount of money raised by the three for charity in all is 386.81
Step-by-step explanation:
Alexis raises 75.23 for charity.
Sue raises 3 times as much as Alexis. This means that the total amount of money raised by Sue is 3 × 75.23= 225.69
The amount of money that Manuel raises is 85.89.
The amount of money raised by all of them(Alexis, Sue and Manuel) would be sum of the amount of money raised by Alexis + the amount of money raised by Sue +
the amount of money raised by Manuel. This becomes
75.23 + 225.69 + 85.89 = 386.81
Answer: 386.81
Step-by-step explanation:
Alexia raises 75.23 = x
Sue raises 3 times Alexia = 3x
= 3 × 75.23
=225.69
Manuel raises 85.89
Total amount raised = Alexia + sue + Manuel
= 75.23 + 225.69 + 85.89
= 386.81
A farmer needs to enclose three sides of a field with a fence (the fourth side is a river). The farmer has 49 yards of fence and wants the field to have an area of 294 sq-yards. What should the dimensions of the field be? (For the purpose of this problem, the width will be the smaller dimension (needing two sides); the length with be the longer dimension (needing one side). Additionally, the length should be as long as possible.)
Answer:
The length of the field = 24.5 yards
The width of the field = 12 yards.
Step-by-step explanation:
If "w" is the width, then the length is 49 - 2w.
The area of the rectangle field = length × width
= w(49 - 2w)
Area = [tex]49w - 2w^2[/tex]
This a quadratic equation, the vertex of x coordinate is w
w = [tex]\frac{-b}{2a}[/tex]
Here a = -2 and b = 49
w = [tex]\frac{-49}{2(-2)} = \frac{-49}{-4} = 12.25[/tex].
So width of the field is 12.25 yards.
The length of the filed = 49 - 2(12.25) = 24.5
Area = 12.25 × 24.5 = 300.125 square yards.
The field has an area of 294.
Therefore, the length must be 24.5 yards and width must be 12 yards.
24.5 × 12 =294 square yards.
Therefore, the length of the field = 24.5 yards
the width of the field = 12 yards.
To find the dimensions of the field, we can use equations based on the area and perimeter. We set up equations for the area and perimeter and solve them simultaneously to find the values of width and length.
Explanation:To find the dimensions of the field, we can use the formula for finding the area of a rectangle: length x width. Let's assume the length of the field is x yards. Since the width is the smaller dimension and requires two sides, we'll represent it as 2w yards. Given that the area of the field is 294 sq-yards, we have the equation x * 2w = 294.
Additionally, we know that the farmer has 49 yards of fence, and he needs to enclose three sides of the field. The three sides are two sides of width (2w) and one side of length (x). So, the total length of the fence needed is 2w + 2w + x = 49.
Simplifying the equation, we have 4w + x = 49. Substituting x from the first equation into the second equation, we get 4w + (294 / 2w) = 49.
Now, we can solve this equation to find the value of w. Once we have the value of w, we can substitute it back into the first equation to find the value of x.
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On the basis of data collected during an experiment, a biologist found that the growth of a fruit fly population (Drosophila) with a limited food supply could be approximated by
N(t) = 600/1+39e^-0.16t
(a) What was the initial fruit fly population in the experiment? where t denotes the number of days since the beginning of the experiment.
(b) What was the population of the fruit fly colony on the t = 11 day? (Round your answer to the nearest integer.)
Answer:
(a). 15
(b). 78
Step-by-step explanation:
Growth of the population of a fruit fly is modeled by
N(t) = [tex]\frac{600}{1+39e^{-0.16t} }[/tex]
where t = number of days from the beginning of the experiment.
(a). For t = 0 [Initial population]
N(0) = [tex]\frac{600}{1+39e^{-0.16\times 0} }[/tex]
= [tex]\frac{600}{1+39}[/tex]
= [tex]\frac{600}{40}[/tex]
= 15
Initial population of the fruit flies were 15.
(b).Population of the fruit fly colony on 11th day.
N(11) = [tex]\frac{600}{1+39e^{-0.16\times 11} }[/tex]
= [tex]\frac{600}{1+39e^{-1.76} }[/tex]
= [tex]\frac{600}{1+39\times 0.172 }[/tex]
= [tex]\frac{600}{1+6.71}[/tex]
= [tex]\frac{600}{7.71}[/tex]
= 77.82
≈ 78
On 11th day number of fruit flies colony were 78.
Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machines Y. AT these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets.
A. 4
B. 6
C. 8
D. 10
E. 12
Answer:
E. 12 days
Step-by-step explanation:
So first, we need to find the rates at which each machine will produce w widgets. For machine y, its rate would be:
[tex]y=\frac{W}{T}[/tex]
where W is the number of widgets produced and T is the time it takes to produce them.
We know that x takes 2 more days to produce the same amount of widgets, so the time it takes machine x to produce them can be written as T+2. This will give us the following rate for machine x:
[tex]x=\frac{W}{T+2}[/tex]
the problem also tells us that the two machines working together will produce 5W/4 widgets in 3 days, so if we add the rates for x and y, we will get the total rate which would be:
[tex]x+y=\frac{5W/4}{3}[/tex]
which can be simplified to:
[tex]x+y=\frac{5W}{12}[/tex]
we can now substitute the rates for x and y in the equation so we get:
[tex]\frac{W}{T+2}+\frac{W}{T}=\frac{5W}{12}[/tex]
we can simplify this equation by dividing everything into W, so we get:
[tex]\frac{1}{T+2}+\frac{1}{T}=\frac{5}{12}[/tex]
and we can multiply everything by the LCD. In this case the LCD is 12T(T+2) so we get:
[tex]\frac{1}{T+2}(12T)(T+2)+\frac{1}{T}(12T)(T+2)=\frac{5}{12}(12T)(T+2)[/tex]
which simplifies to:
12T+12(T+2)=5T(T+2)
we can do the respective multiplications so we get:
[tex]12T+12T+24=5T^{2}+10T[/tex]
which simplifies to:
[tex]24T+24=5T^{2}+10T[/tex]
and now we can set the equation equal to zero so we end up with:
[tex]5T^{2}+10T-24T-24=0[/tex]
which simplifies to:
[tex]5t^{2}+14T-24=0[/tex]
now we can solve this by any of the available methods there are to solve quadratic equations. I will solve it by factoring, so we get:
(5T+6)(T-4)=0
so we can set each of the factors equal to zero so we get:
5T+6=0
[tex]T=-\frac{6}{5}[/tex]
this answer isn't valid because there is no such thing as a negative time. So we find the next time then:
T-4=0
T=4
So it takes 4 days for machine x to produce W widgets. We can now rewrite x's rate like this:
[tex]x=\frac{W}{T+2}[/tex]
so
[tex]x=\frac{W}{4+2}[/tex]
[tex]x=\frac{W}{6}[/tex]
With this information, we know that the number of wigets produced can be found by using the following formula:
W=xd
in this case d is the number of days (this is for us not to confuse the previous T with the new time)
so when solving for d we get that:
[tex]d=\frac{W}{x}[/tex]
so when substituting we get that:
[tex]d=\frac{2W}{W/6}[/tex]
when simplifying we get that:
d=12
A solid lies between planes perpendicular to the y-axis at yequals0 and yequals2. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola x equals StartRoot 6 EndRoot y squared. Find the volume of the solid.
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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Why is the answer E?
Answer:
E
Step-by-step explanation:
For a function to be differentiable at a point, it must be continuous at that point [ f(x⁻) = f(x⁺) ], and smooth at that point [ f'(x⁻) = f'(x⁺) ].
f(-1⁻) = 3(-1) + 5 = 2
f(-1⁺) = -(-1)² + 3 = 2
So the function is continuous.
f'(-1⁻) = 3
f'(-1⁺) = -2(-1) = 2
So the function is not smooth.
Therefore, the derivative f'(-1) does not exist.
Which are the solutions of x2 = –13x – 4? 0, 13 0, –13 StartFraction 13 minus StartRoot 153 EndRoot Over 2 EndFraction comma StartFraction 13 + StartRoot 153 EndRoot Over 2 EndFraction StartFraction negative 13 minus StartRoot 153 EndRoot Over 2 EndFraction comma StartFraction negative 13 + StartRoot 153 EndRoot Over 2 EndFraction
Answer:
[tex]x_{1}=\frac{-13+\sqrt{153}}{2}\\x_{2}=\frac{-13-\sqrt{153}}{2}[/tex]
Step-by-step explanation:
The given expression is
[tex]x^{2}=-13x-4[/tex]
To solve this quadratic equation, we first need to place all terms in one side of the equation sign
[tex]x^{2} +13x+4=0[/tex]
Now, to find all solutions of this expression, we have to use the quadratic formula
[tex]x_{1,2}=\frac{-b\±\sqrt{b^{2}-4ac}}{2a}[/tex]
Where [tex]a=1[/tex], [tex]b=13[/tex] and [tex]c=4[/tex]
Replacing these values in the formula, we have
[tex]x_{1,2}=\frac{-13\±\sqrt{(13)^{2}-4(1)(4)}}{2(1)}\\x_{1,2}=\frac{-13\±\sqrt{169-16}}{2}=\frac{-13\±\sqrt{153}}{2}[/tex]
So, the solutions are
[tex]x_{1}=\frac{-13+\sqrt{153}}{2}\\x_{2}=\frac{-13-\sqrt{153}}{2}[/tex]
If we approximate each solution, it would be
[tex]x_{1}=\frac{-13+\sqrt{153}}{2}\approx -0.32\\\\x_{2}=\frac{-13-\sqrt{153}}{2} \approx -12.68[/tex]
Answer:
D on Edge
Step-by-step explanation:
The goals against average (A) for a professional hockey goalie is determined using the formula A = 60 . In the formula, g represents the number of goals scored against the goalie and t represents the time played, in minutes. Which is an equivalent equation solved for g?
A_At/60 = g
B_ A/60t= g
C_60A/t = g
D_60At = g
Answer:
A. [tex]\frac{At}{60}=g[/tex]
Step-by-step explanation:
Consider the formula [tex]A = 60(\frac{g}{t})[/tex] is given,
We have to find : The formula for g.
For this we need to isolate g in one side of the equation,
Multiply both sides of the equation by t, ( using multiplicative property of equality )
We get,
[tex]At = 60g[/tex]
Divide both sides by 60 ( division property of equality )
We get,
[tex]\frac{At}{60}=g[/tex]
Which is the required equivalent equation.
That is,
OPTION A would be correct.
Answer: A. At/60 = g
Step-by-step explanation: Consider the formula is given,
We have to find : The formula for g.
For this we need to isolate g in one side of the equation,
Multiply both sides of the equation by t, ( using multiplicative property of equality )
We get,
Divide both sides by 60 ( division property of equality )
We get,
Which is the required equivalent equation.
That is,
OPTION A would be correct.
In a study of speed dating, male subjects were asked to rate the attractiveness of their female dates, and a sample of the results is listed below (1equalsnot attractive; 10equalsextremely attractive). Construct a confidence interval using a 99% confidence level. What do the results tell about the mean attractiveness ratings of the population of all adult females? 6, 9, 3, 9, 6, 6, 7, 7, 8, 9, 3, 8
Answer:
The 99% confidence interval is be given by (4.872;8.628)
Step-by-step explanation:
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The dataset is:
6, 9, 3, 9, 6, 6, 7, 7, 8, 9, 3, 8
2) Compute the sample mean and sample standard deviation.
In order to calculate the mean and the sample deviation we need to have on mind the following formulas:
[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex]
The value obtained is [tex]\bar X=6.75[/tex]
[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}[/tex]
The sample deviation obtained is [tex]s=2.094[/tex]
3) Find the critical value t* Use the formula for a CI to find upper and lower endpoints
In order to find the critical value we need to take in count that our sample size n =12 <30 and on this case we don't know about the population standard deviation, so on this case we need to use the t distribution. Since our interval is at 99% of confidence, our significance level would be given by [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2 =0.005[/tex]. The degrees of freedom are given by:
[tex]df=n-1=12-1=11[/tex]
We can find the critical values in excel using the following formulas:
"=T.INV(0.005,11)" for [tex]t_{\alpha/2}=-3.106[/tex]
"=T.INV(1-0.005,11)" for [tex]t_{1-\alpha/2}=3.106[/tex]
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]
The next step would be calculate the limits for the interval
Lower interval :
[tex]\bar X - t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]
[tex]6.75 - 3.106 \frac{2.094}{\sqrt{12}}=4.872[/tex]
Upper interval :
[tex]6.75 + 3.106 \frac{2.094}{\sqrt{12}}=8.628[/tex]
So the 99% confidence interval would be given by (4.872;8.628)
99% of the time, when we calculate a confidence interval with a sample of n=12, the true mean of rate of attractiveness of their female dates will be between the 4.872 and 8.628.
The vertical line passing through the vertex of a parabola is called the
Answer:
Axis of symmetry.
Step-by-step explanation:
We have been given an incomplete statement. We are supposed to complete the given statement.
Given statement: The vertical line passing through the vertex of a parabola is called the ________.
We know that a parabola is symmetric about axis of symmetry . The line passing through the vertex of parabola divides the parabola into two mirror images.
Therefore, the vertical line passing through the vertex of a parabola is called the the axis of symmetry.
Upper A 55 ft wire is used to brace a utility pole. If the wire is attached 6 ft from the top of the 40 ft pole, how far from the base of the pole will the wire be attached to the ground
Listed below are measured amounts of caffeine? obtained in one can from each of 14 brands. Find the? range, variance, and standard deviation for the given sample data. Include appropriate units in the results [mg per 12oz drink; (mg per 12oz of drink)2; brands2; brands]. Are the statistics representative of the population of all cans of the same 14 brands? consumed?
31
52
35
57
0
32
35
52
46
41
30
41
0
0
Answer:
Range is 57.
Variance is 375.143.
standard deviation is 19.37.
Step-by-step explanation:
Consider the provided information.
Range is the difference between highest and lowest data value.
The highest data value is 57 and lowest is 0.
Thus the range is 57-0=57
Range is 57.
Mean is the sum of data value divided by the number of data value:
[tex]\bar x=\frac{31+52+35+57+0+32+35+52+46+41+30+41+0+0}{14}\approx32.286[/tex]
The variance is the sum of squared deviation from the mean divided by n-1.
[tex]s^2 =\frac{\sum(x_i -\bar x)^2}{n - 1}[/tex]
Substitute the respective values in the above formula we get:
[tex]s^2=\frac{(31 - 32.286)^2 +(52-32.286)^2+ ... + (0 -32.286)^2}{14 - 1}\approx 375.143[/tex]
Hence, the variance is 375.143.
Standard deviation is square root of variance.
standard deviation = [tex]\sqrt{375.143}[/tex]
standard deviation ≈ 19.37
Hence, standard deviation is 19.37.
For all cans consumed, the statistics are not representative of the population because in the calculations each brand is weighted equally. Each of the 14 brands of soda is unlikely to be consumed in the same way.
It is very unlikely that all 14 drinks are consumed equally. So,given data is not representative of population
Julie rides her bike from the sports complex to the school. Then she rides from the school to the mall, and then on to the library. Kyle rides his bike from his house to the mall, and then to the library.
The person that traveled the most distance is Julie.
What is an expression?
An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
From the figure,
Julie:
Total distance covered.
= sports complex to school + school to mall + mall + library
= 2/3 + 2/5 + 1(1/3)
= 2/3 + 2/5 + 4/3
= (10 + 6 + 20)/15
= 36/15
= 12/5
= 2(2/5) miles
= 2.4 miles
Kyle:
Total distance covered.
= house to mall + mall to library
= 4/5 + 1(1/3)
= 4/5 + 4/3
= (12 + 20)/15
= 32/15
= 2(2/15) miles
= 2.13 miles
Thus,
Julie has traveled more distance than Kyle.
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A rain storm came through Clifton park and it was accumulating 2/3 inches of rain/hour. How many inches of rain would fall in 6 hours if it continued at this rate?
4 inches of rain would fall in 6 hours
Solution:Given that, A rain storm came through Clifton park
And it was accumulating [tex]\frac{2}{3}[/tex] inches of rain/hour
So amount of rain accumulated in 1 hour = [tex]\frac{2}{3}[/tex]
Thus amount of rain accumulated in six hours is calculated by multiplying the amount of water accumulating per hour and 6
Amount of water accumulated in 6 hours = Amount of water accumulated in 1 hour [tex]\times[/tex] 6
[tex]\text { Amount of water accumaulated in 6 hours }=\frac{2}{3} \times 6=4[/tex]
Another way:
Let "n" be the amount of rain accumulated in 6 hours
1 hour ⇒ [tex]\frac{2}{3}[/tex] rain accumulated
6 hours ⇒ "n"
By cross multiplication, we get
[tex]6 \times \frac{2}{3} = 1 \times n\\\\n = \frac{2}{3} \times 6 = 4[/tex]
Hence, 4 inches of rain would fall in 6 hours.
If the rain fell at a constant rate of 2/3 inch per hour, then 4 inches of rain would fall in a total of 6 hours. This calculation is made by multiplying the rate of rainfall by the total time.
Explanation:The question asks how many inches of rain would fall in Clifton park in 6 hours if the rate was consistently 2/3 inch per hour. Given the constant rate of rainfall, we can calculate the total inches of rain that fell in 6 hours by multiplying the rate (2/3 inches/hour) by the total time in hours (6 hours).
So, doing the multiplication:
(2/3 inch/hour) * (6 hours) = 4 inches of rain.
This means that if the rain continued to fall at the same rate, we would expect 4 inches of rain to accumulate in Clifton park over 6 hours.
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By recognizing the series below as a Taylor series evaluated at a particular value of x, find the exact sum of the convergent series. 1 + 3/1! + 9/2! + 27/3! + 81/4! + ... + 3n/n! +.......
Answer:
[tex]e^3[/tex]
Step-by-step explanation:
Given is a series as
[tex]1+\frac{3}{1!} +\frac{3^2}{2!} +...+\frac{3^n}{n!} +...[/tex]
Recall the expansion of
[tex]e^x = 1+x+\frac{x^2}{2!} +...+\frac{x^n}{n!} +...[/tex]
This expansion is valid for all real values of x.
Comparing this with our series we find that x =3
Hence the given series =[tex]e^3[/tex]
Thus we find that the given series can be recognized with the expansion of exponential series with powers of e and here we see that power of e is 3.
So the given Taylor series is equivalent to
[tex]e^3[/tex]
The series in the question is a Taylor series representing e^3x. At x=1, the sum of the series is exactly e^3.
Explanation:The series you provided can be recognized as a Taylor series, an infinite sum of terms calculated from the values of a function's derivatives at a single point. Specifically, it resembles the Taylor series representation of the exponential function ex, which is 1 + x/1! + x2/2! + x3/3! + ... and so on.
Looking at your series 1 + 3/1! + 9/2! + 27/3! + 81/4! ..., we can see that each term 3n/n! is equivalent to (3n)/n!, which can be rewritten as (3^n)(1/n!). This yields a series in the form of 1 + 3x/1! + (3x)2/2! + (3x)3/3! + ... It's apparent that your series can be rewritten as e3x. So when x = 1, the sum of the series is e3, which is exactly the number e cubed.
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Simplify
(4x−3+2x2)+(2x+1)
a.8x2−2
b.2x2+2x+4
c.2x2+6x−2
d.2x2−6x−2
Aa. 8x2 - 2
i
Step-by-step explanation:
4x2 = 8
8-3 = 5
5 + 4 =
Answer:
c. 2x2+6x−2
Step-by-step explanation:
Arica can run 1/6 of a kilometer in a minute her school is 3/4 A kilometer away from her home At this speed how long will it take Erica to run from home to school
Erica will take 4.5 minutes to run from home to school
Solution:Given that , Arica can run [tex]\frac{1}{6}[/tex] of a kilometer in a minute
Her school is [tex]\frac{3}{4}[/tex] th of a kilometer away from her home
We have to find at this speed how long will it take Erica to run from home to school
The relation between speed distance and time is given as:
[tex]\text { Distance }=\text { speed } \times \text { time }[/tex]
Plugging in values, we get
[tex]\frac{3}{4}=\frac{1}{6} \times \text { time taken }[/tex]
[tex]\begin{array}{l}{\text { Time taken to reach school }=\frac{3}{4} \times 6} \\\\ {\text { Time taken to reach home }=\frac{3}{2} \times 3} \\\\ {\text { Time taken to reach home }=\frac{9}{2}=4.5}\end{array}[/tex]
Hence, she takes 4.5 minutes to reach school from her home
Write the ratios for sin A and cos A
Answer: The correct option is C
Step-by-step explanation:
Looking at the right angle triangle ABC, three sides are known and the angles are unknown. To find sin A, we will take A to be our reference angle, we will have the following
Hypotenuse = AB = 26
Opposite side = BC = 24
Adjacent side = AC = 10
Applying trigonometric ratio
SinA = opposite/hypotenuse
SineA = 24/26
To find cos A, A remains our reference angle, we will have the following
Hypotenuse = AB = 26
Opposite side = BC = 24
Adjacent side = AC = 10
Applying trigonometric ratio
CosA = adjacent/hypotenuse
CosA = 10/26
The correct option is C
Which of the following sums does not equal the others?
the sum from i equals 1 to 3 of i squared
the sum from i equals 1 to 2 of i cubed
the sum from i equals 1 to 4 of the quantity i plus 1
the sum from i equals 4 to 5 of the quantity 2 times i minus 2
Sum of Option 2 does not equal to others
Step-by-step explanation:
We have to find each sum to check which is a outlier.
so,
Option 1:
∑i^2 where i = 1 to 3
[tex]Sum = 1^2+2^2+3^2\\=1+4+9\\=14[/tex]
Option 2:
∑i^3 where i = 1 to 2
So,
[tex]Sum = 1^3+2^3\\= 1 +8\\=9[/tex]
Option 3:
∑(i+1) where i = 1 to 4
[tex]Sum = (1+1) + (2+1) +(3+1) +(4+1)\\=2+3+4+5\\=14[/tex]
Option 4:
∑(2i-2) where i = 4 to 5
[tex]Sum = [2(4)-2]+[2(5)-2]\\=(8-2)+(10-2)\\=6+8\\=14[/tex]
Hence,
Sum of Option 2 does not equal to others
Keywords: Sum, Formulas
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The sum from i equals 1 to 2 of i cubed does not equal the others.
Explanation:To determine which sum does not equal the others, we need to evaluate each sum.
The sum from i equals 1 to 3 of i squared: 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14The sum from i equals 1 to 2 of i cubed: 1^3 + 2^3 = 1 + 8 = 9The sum from i equals 1 to 4 of the quantity i plus 1: (1+1) + (2+1) + (3+1) + (4+1) = 2 + 3 + 4 + 5 = 14The sum from i equals 4 to 5 of the quantity 2 times i minus 2: (2*4-2) + (2*5-2) = 6 + 8 = 14From the evaluations, we can see that the sum from i equals 1 to 2 of i cubed does not equal the others.
Anna is an avid reader. Her generous grandparents gave her money for her birthday, and she decided to spend at most $150.00 on books. Reading Spot is running a special: all paperback books are $8.00 and hardback books are $12.00. Anna wants to purchase at least 12 books.
1.) Write a system of inequalities to reach po represent the situation.
2.) Graph the region of the solutions to the inequality.
3.) Name two different solutions for Anna's situation.
Answer:
The solutions for 3 questions are explained one after the other below.
Step-by-step explanation:
1).Let x be the number of paperback books that she buys, y be the number of hardback books that she buys.
for the first condition, i.e, she has decided to spend at most $150.00 on books,the required inequality will be :
[tex]8x+12y\leq 150[/tex]
for the second condition , i.e, she wants to purchase at least 12 books,
the required inequality will be:
[tex]x+y\geq 12[/tex]
2). the graph is in the attachment..
3). x,y are the two required solutions. where,
x =number of paperback books she buys.
y=number of hardback books she buys.
Answer:
1) equations 1, 2, 3 and 4
2) see picture attached (the region of the solutions is in yellow)
3) x = 18.75 and y =0
y = 12.5 and x =0
Step-by-step explanation:
Let's call x the number of paperback books bought and y the number of hardback books bought.
She decided to spend at most $150.00. All paperback books are $8.00 and all hardback books are $12.00. Combining this information we get:
x*8 + y*12 ≤ 150 (eq. 1)
Anna wants to purchase at least 12 books. Mathematically:
x + y ≥ 12 (eq. 2)
On the other hand, both the number of paperback books bought and the number of hardback books bought must be positive, that is:
x ≥ 0 (eq. 3)
y ≥ 0 (eq. 4)
3) One possible solution is got if we make y = 0 and to take eq. 1 as an equality, then:
From eq. 1: x*8 = 150
x = 150/8 = 18.75
equations 2 and 3 are also satisfied
Another option is to make x = 0 and to take eq. 1 as an equality, then:
From eq. 1: y*12 = 150
y = 150/12 = 12.5
equations 2 and 4 are also satisfied
Please help will mark brainliest!!!
Answer:
y = 6
Step-by-step explanation:
Its going by 6's
If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli's Law gives the volume V of water remaining in the tank after t minutes as V=5000(1−t40)20≤t≤40. Find the rate at which water is draining from the tank after the following amount of time. (Remember that the rate must be negative because the amount of water in the tank is decreasing.)
Answer:
V'(t) = [tex]-250(1 - \frac{1}{40}t)[/tex]
If we know the time, we can plug in the value for "t" in the above derivative and find how much water drained for the given point of t.
Step-by-step explanation:
Given:
V = [tex]5000(1 - \frac{1}{40}t )^2[/tex] , where 0≤t≤40.
Here we have to find the derivative with respect to "t"
We have to use the chain rule to find the derivative.
V'(t) = [tex]2(5000)(1 - \frac{1}{40} t)d/dt (1 - \frac{1}{40}t )[/tex]
V'(t) = [tex]2(5000)(1 - \frac{1}{40} t)(-\frac{1}{40} )[/tex]
When we simplify the above, we get
V'(t) = [tex]-250(1 - \frac{1}{40}t)[/tex]
If we know the time, we can plug in the value for "t" and find how much water drained for the given point of t.
(25 POINTS) PLEASE HELP WILL GIVE BRAINLIEST, THANKS AND 5 STAR RATING!!!
5 QUESTIONS SHOW WORK!
Answer:
Step-by-step explanation:
Solve the equation by first using a Sum-to-Product Formula. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate.) sin(5θ) − sin(3θ) = cos(4θ)
Answer:
Solutions of the equation are 22.5°, 30°.
Step-by-step explanation:
The given equation is sin(5θ) - sin(3θ) = cos(4θ)
We take left side of the equation
sin(5θ) - sin(3θ) = [tex]2cos(\frac{5\theta+3\theta}{2})sin(\frac{5\theta-3\theta}{2})[/tex]
= [tex]2cos(4\theta)sin(\theta)[/tex] [From sum-product identity]
Now we can write the equation as
2cos(4θ)sin(θ) = cos(4θ)
2cos(4θ)sinθ - cos(4θ) = 0
cos(4θ)[2sinθ - 1] = 0
cos(4θ) = 0
4θ = 90°
θ = [tex]\frac{90}{4}[/tex]
θ = 22.5°
and (2sinθ - 1) = 0
sinθ = [tex]\frac{1}{2}[/tex]
θ = 30°
Therefore, solutions of the equation are 22.5°, 30°
a fish tank in the shape of a rectangular prism measures 100 x 60 x 40/The water level reached the midpost of the base (50cm mark) when the tank was tilted to rest on a 60cm edge. What would be the depth of the water, if the tank is returned to its horizontal position (resting on a 60x100 base)?
Answer:
Step-by-step explanation:
volume of water
[tex]=\frac{1}{2}*50*40*60=60000 ~cm^3[/tex]
when the base is 100×60
let h be depth of water.
100×60×h=60000
h=60000/6000=10 cm.
What is the product? 2y/y-3 x 4y -12/2y+8
Answer:
4y/(y+4)
Step-by-step explanation:
2y/(y-3) x [(4y -12) /(2y+8)]
To determine this, at first we have to break the parentheses. Since there is no matching values, we have to multiply the numerators and denominators.
[2y x (4y - 12)] / (y-3) x (2y + 8)
or, [(2y*4y) - (2y*12)]/[(y*2y) + (y*8) - (3*2y) - (3*8)]
(using algebraic equation)
or, (8y^2 - 24y)/(2y^2 + 8y - 6y - 24)
or, (8y^2 - 24y)/(2y^2 + 2y - 24)
or, 8y(y - 3)/2(y^2 + y - 12) (taking common)
or, 4y(y - 3)/(y^2 + 4y - 3y - 12)
or, 4y(y - 3)/[y(y + 4) - 3 (y + 4)] (Using factorization or Middle-Term factor)
or, 4y (y - 3)/(y + 4)(y - 3)
or, 4y/(y + 4) [as (y-3)/(y-3) = 1, we have dropped the part]
The answer is = 4y/(y+4)
find the range of the given function f(x)=(x-1)^2+1
a.) [1, infinity)
b.) (- infinity, infinity)
c.) [0, infinity)
d.) (- infinity, 1)
Answer:
a.) [1, infinity)
Step-by-step explanation:
The equation is that of a parabola that opens upward with vertex (1, 1). Hence the minimum value of f(x) is 1, and all values greater than that are part of the range: [1, ∞).
_____
The "vertex form" of the equation of a parabola is ...
f(x) = a(x -h)^2 + k
The vertex is at (h, k). When a > 0, the parabola opens upward. When a < 0, the parabola opens downward. Whichever way it opens, the value k is an extreme value and the limit of the range.
This is my last problem on this sample work. Dx Please help! There are no multiple choice options so this is a free for all. Thank you!!
Answer:
[tex]\left\{\begin{array}{l}y\ge 2x+4\\ \\y<-x+2\end{array}\right.[/tex]
Step-by-step explanation:
1. The solid line passes trough the points (0,4) and (-2,0). The equation of this line is:
[tex]\dfrac{x-0}{-2-0}=\dfrac{y-4}{0-4}\\ \\y-4=2x\\ \\y=2x+4[/tex]
The origin doesn't belong to the shaded region, so its coordinates do not satisfy the inequality. Thus,
[tex]y\ge 2x+4[/tex]
2. The dotted line passes trough the points (0,2) and (2,0). The equation of this line is:
[tex]\dfrac{x-0}{2-0}=\dfrac{y-2}{0-2}\\ \\y-2=-x\\ \\y=-x+2[/tex]
The origin belongs to the shaded region, so its coordinates satisfy the inequality. Thus,
[tex]y< -x+2[/tex]
Hence, the system of two inequalities is
Which is the graph of y = ⌊x⌋ – 2?
Answer:
The third graph from left to right
Step-by-step explanation:
The function [tex]f(x)=\left [ x \right ][/tex] is called Greatest Integer Function of x is such that it returns the largest integer less than or equal to x
Some examples of points are (0,0),(0.5,0),(1,1),(1.9,1),(-0.7,-1)
Since our function is
[tex]f(x)=\left [ x \right ]-2[/tex]
We must subtract 2 to the points above like
(0,-2),(0.5,-2),(1,-1),(1.9,-1),(-0.7,-3)
The only graph that complies with such requirements is the third one
when 2x^3-3x^2+kx-1 is divided by x-1 the remainder is 2 find k?
Answer:
k = 4
Step-by-step explanation:
The remainder theorem tells you that the remainder from division of f(x) by (x-1) is f(1). Evaluating the expression for x=1 gives ...
2(1³) -3(1²) +k(1) -1 = 2 -3 +k -1 = k -2
We want this to be equal to 2, so ...
k -2 = 2
k = 4
Applying the Remainder Theorem to the given polynomial, we can substitute x = 1 into the polynomial equation and solve for k, which gives us k = 4.
Explanation:The question asks to find the value of k when given polynomial 2x^3 - 3x^2 + kx - 1 is divided by x - 1 and the remainder is 2. We utilize the Remainder Theorem for this, which states that when a polynomial f(x) is divided by x-c, the remainder is equal to f(c).
So, by substituting x = 1 in the given polynomial as per the Remainder Theorem, we have: 2(1)^3 - 3(1)^2 + k(1) - 1 = 2. Simplifying this equation leads us to: 2 - 3 + k -1 = 2, which can further be simplified to k - 2 = 2. Thereby, solving for k gives us k = 4.
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