Of all rectangles with a perimeter of 29​, which one has the maximum​ area? (Give the​ dimensions.) Let A be the area of the rectangle. What is the objective function in terms of the width of the​ rectangle, w? Aequals 14.5 w minus w squared ​(Type an​ expression.) The interval of interest of the objective function is nothing. ​(Type your answer in interval notation. Use integers or simplified fractions for any numbers in the​ expression.) The rectangle that has the maximum area has length nothing and width nothing. ​(Simplify your​ answer.)

Answers

Answer 1

Answer:

For maximum area: Its a square of dimensions 7.25 * 7.25.

Step-by-step explanation:

We can use calculus to solve this:

Let the length of the rectangle be L and the width be W then we have

2L + 2W = 29

L + W = 14.5

W = 14.5 - L

Area = WL

A = L(14.5 - L)

A = 14.5L - L^2

We need this to be a maximum.

Finding the derivative:

dA/dW = 14.5 - 2L = 0   for a maximum

2L = 14.5

L = 7.25

Also W = 14.5 - L

= 14.5 - 7.25 = 7.25.

So the dimensions for maximum area are 7.25 * 7.25.

The  figure is a SQUARE.

Answer 2

The perimeter of a rectangle, is the sum of its side lengths.

The rectangle that has the maximum area when the length is 7.25 and the width is 7.25

The perimeter is given as:

[tex]\mathbf{P = 29}[/tex]

Let the dimensions be l and w.

So, we have:

[tex]\mathbf{2( l + w) = 29}[/tex]

Divide both sides by 2

[tex]\mathbf{l + w = 14.5}[/tex]

Make l the subject

[tex]\mathbf{l = 14.5 - w}[/tex]

The area of a rectangle is:

[tex]\mathbf{A = lw}[/tex]

Substitute [tex]\mathbf{l = 14.5 - w}[/tex]

[tex]\mathbf{A = (14.5 - w)w}[/tex]

Open brackets

[tex]\mathbf{A = 14.5w - w^2}[/tex]

Differentiate

[tex]\mathbf{A' = 14.5 - 2w}[/tex]

Set to 0

[tex]\mathbf{14.5 - 2w = 0}[/tex]

Collect like terms

[tex]\mathbf{2w = 14.5}[/tex]

Divide both sides by 2

[tex]\mathbf{w = 7.25}[/tex]

Recall that: [tex]\mathbf{l = 14.5 - w}[/tex]

So, we have:

[tex]\mathbf{l = 14.5 - 7.25 = 7.25}[/tex]

Hence, the rectangle that has the maximum area when the length is 7.25 and the width is 7.25

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Related Questions

A small wooden statue has a mass of 500 g and a volume of 670 cubic centimeters. What is the density of the statue?

Answers

Answer:

0.7463 gm/cm^3

Step-by-step explanation:

Density of any substance is mass of that substance per unit of volume.

it can be represented by given below expression

Density = mass/ volume

Given

mass = 500gm

volume = 670 cubic centimetres

Therefore, density = 500/670 gm/cubic centimetres

density = 0.7463 gm/cm^3

Thus , density of statue is 0.7463 gm/cm^3.

How many different 12-letter arrangements can be made using the letters in the word indianapolis?

Answers

Answer:

764

Step-by-step explanation:

If the first term of an arithmetic sequence is 4 and the difference between con-
secutive terms is 5, what is the second term? The third term? The twelfth? The
eightieth? The nth?

Answers

Answer:third term=14 twelfth term=59

Step-by-step explanation:

first term=a=4

Common difference=d=5

Second term=4+5=9

Third term=9+5=14

Using Tn=a+d(n-1)

T12=4+5(12-1)

T12=4+5x11

T12=4+55

T12=59

Twelveth term=59

please help this is timed please!

Answers

Answer:

f/g(x) 1/x x=0

Step-by-step explanation:

f/g 1/x+2

x+2=0

-2      -2

x=-2

In Spanish club, the boy-girl ratio is 8-9. if 72 members are girls, how many are boys?

Answers

Answer:

The number of boys are 64.

Step-by-step explanation:

We have,

The ratio of the boy to girl is 8:9.

Total number of girls are 72.

Let total boys are 8x and total girls are 9x.

According to the question,

[tex]9x=72\\\\x=8[/tex]

Since, number of boys are 8x. So,

[tex]B=8x=8\times 8=64[/tex]

So, the number of boys are 64.

Giving brainliest to CORRECT answer.

Answers

Answer:

D

If you put the equation into a calculator, it will show a graph of it.

When plotting points on the coordinate plane below, which point would lie on the y-axis?
(0, 1)
(7, 0)
(6, 8)
(8, 2)

Answers

1, 0 , 8, and 2 would all lie on the y-axis

Answer:

(0,1)

Step-by-step explanation:

i got it right

For a sphere with a radius of 8, what is the surface area

Answers

Answer:

64

Step-by-step explanation:

There were 15 students running in a race. How many different arrangements of first, second, and third place are possible?

Answers

Answer:

The area 2730 possible arrangements of first, second, and third place.

Step-by-step explanation:

The order in which the students finish is important. For example A, B and C is a different finishing order(outcome) than B, A and C. So we use the permutations formula to solve this question.

Permutations formula:

The number of possible permutations of x elements from a set of n elements is given by the following formula:

[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]

In this question:

3 from a set of 15. So

[tex]P_{(15,3)} = \frac{15!}{(15-3)!} = 2730[/tex]

The area 2730 possible arrangements of first, second, and third place.

An equilateral triangle has what type of symmetry?

Answers

Answer:

An equilateral triangle has 3 lines of symmetry

Hope this helps

Step-by-step explanation:

Which types of triangles can be formed by taking the cross-section of a rectangular prism like the one shown above?

Answers

Answer: Equilateral, isosceles and scalene.

Step-by-step explanation:

ivana has 5.50 luc has 1.4 times as much as ivana, how many indian rupees would ivana get if the value of rupees are 61.92 per dollar rate

Answers

i think it would be 24.67 dollars because thats what i calculated but i dont know

Select the closest answer. *
20 points
1. Find the surface area of a sphere with a radius of 4.2 centimeters.
A. 235.6 cm?
C. 121.4 cm
B. 310.3 cm
D. 221.7 cm
A
B
С
OD
Select the closest answer.

Answers

Answer:

D. 221.7 cm

Step-by-step explanation:

Surface area of a sphere is: [tex]SA=4\pi r^2[/tex]

'r' - radius

We are given the radius of 4.2 centimetres.

[tex]SA=4\pi 4.2^2\\4 * \pi * 4.2^2\\\rightarrow 4.2^2 =17.64\\\text {Using 3.14 for pi: }\\4 * 3.14 *17.64\\12.56*17.64\\\boxed {221.5584}[/tex]

221.5584 ≈ 221.6

The closest answer is Option D, therefore it should be the correct answer.

Complete 12 for 10 points.

Answers

Answer:

5 minutes

Step-by-step explanation:

200+15=215

214/43=5

It moved up a total of 200 + 15 = 215 meters at 43 meters per minute

t = 215 m / (43 m/min) = 5 minutes

Answer: 5 minutes

very easy question please answer asap giving brainliest

Answers

Answer:

D understand the problem.

Step-by-step explanation:

Answer:

gathering of resources is the main first thing .....

4m 5m what is the value of x

Answers

Answer:

5m x 5m, 6m x 4m and 7m x 4m

Step-by-step explanation:

in interval of 0.02 and for the same value of m.

tell me the answer to the question plz​

Answers

Step-by-step explanation:

(11.99+23.98+9.95)= 45.92

she has to pay = 45.92 - 20/100 (45.92)

= 45.92 - 9.18 = 36.74

so option

about 37 is right

P
Which expression is equal to 2 x 15?

Answers

Answer:

30

Step-by-step explanation:

2*15

= 30

Answer:

30

Step-by-step explanation:

2*15

= 30

What value of x is in the solution set of 4x - 123 16 + 8x?

Answers

Answer:

the value of x is in the solution set of 4x-12<16=8x

Step-by-step explanation:

multiply the 4x take away from 12= 8x

Please help ASAP! I will mark BRAINLIEST! Please answer CORRECTLY! No guessing!

Answers

Answer:

Graph B

Step-by-step explanation:

Find the values of the variable y, for which:
the difference between the fractions y+12/y-4 and y/y+4 is equal to their product


Answers

Answer:

  y = 12

Step-by-step explanation:

Ordinarily, in math, when we say "the difference between ..." we mean the second listed item is subtracted from the first. (In English, we often mean "the positive difference between ..." so the order doesn't matter.)

We'll assume we are only interested in the case where the difference is in the order listed.

  [tex]\dfrac{y+12}{y-4}-\dfrac{y}{y+4}=\dfrac{y+12}{y-4}\cdot\dfrac{y}{y+4}\\\\\dfrac{(y+12)(y+4)-(y-4)(y)}{y^2-16}=\dfrac{(y+12)(y)}{y^2-16}\\\\0=\dfrac{y^2-8y-48}{y^2-16}=\dfrac{(y-12)(y+4)}{(y-4)(y+4)}=\dfrac{y-12}{y-4}\qquad\text{$y\ne -4$}\\\\\boxed{y=12}[/tex]

There is one solution.

_____

Comment on the solution

If you were to "clear fractions" by multiplying the equation by (y^2-16), you would also obtain the extraneous solution y=-4. That is why we did not solve it that way.

Two angles are supplementary. The measure of the larger angle is 12 degrees more than three times the smaller angle. Find the measures of the angles

Answers

Answer:

Measure of first angle: 42 degrees,

Measure of second angle: 138 degrees

Step-by-step explanation:

Let us say x ⇒ measure of the smaller angle, y ⇒ measure of the larger angle

1. Now given that the measure of the larger angle is 12° more than 3 times the smaller angle: y = 12 + 3x

2. If these angles are supplementary, we could formulate another equation were the two angles add to 180°: x + y = 180

3. In this situation it is convenient to apply substitution as to solve for x, y:

1. x + ( 12 + 3x ) = 180

2. x + 12 + 3x = 180

3. 4x + 12 + 180

4. 4x = 168

5. x = 42 degrees

4. Now substitute this value of x into the first equation y = 12 + 3x, and to solve for y as well:

1. y = 12 + 3 ( 42 )

2. y = 12 + 126

3. y = 138 degrees

Here we need to find two angles given that the angles are supplementary and one is 12° larger than the other.

the two angles are 84° and 96°.

We should start by defining what supplementary angles are.

Two angles a and b are supplementary if:

a + b = 180°.

Now, we also know that one of the angles is 12° larger than the other, so we can write:

a = b + 12°

Now that we have that equation, we can replace it in the first one, so we get:

a + b = 180°

(b + 12°) + b = 180°

Now we can solve this for b.

2*b + 12° = 180°

2*b = 180° - 12°  = 168°

b = 168°/2 = 84°

To find the value of a, we can use the equation:

a = b + 12°

a = 84° + 12° = 96°.

Then the two angles are:

84° and 96°.

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The design for a playground is represented on the coordinate ​plane below, where the units are in yards.

​A fence will be built ​around the perimeter of the entire playground.

​How many yards ​of fencing will be needed to build the fence?

Answers

Answer:

The perimeter of the field is 21 yards

Step-by-step explanation:

See attachment 1 for the coordinates plane.

The question is best solved using an attachment.

To calculate the perimeter of the fence, we simply add the sides of the rectangle.

Attachment 2 is a labelled diagram of attachment 1.

From 2, we have that

Perimeter = A + B + C + D + E + F

Where A to F represent the length of each sides of the fence

From coordinates geometry,

A = 5

B = 4

C = 6½

D = 1½

E = 1½

F = 2½

So,

Perimeter = 5 + 4 + 6½ + 1½ + 1½ + 2½

Perimeter = 21

Hence, the perimeter of the field is 21 yards

Harris Interactive® conducted a poll of American adults in August of 2011 to study the use of online medical information. Of the 1,019 randomly chosen adults, 60% had used the Internet within the past month to obtain medical information. Use the results of this survey to create an approximate 95% confidence interval estimate for the percentage of all American adults who have used the Internet to obtain medical information in the past month.

Answers

Answer:

[tex]0.60 - 1.96\sqrt{\frac{0.60(1-0.60)}{1019}}=0.570[/tex]

[tex]0.60 + 1.96\sqrt{\frac{0.60(1-0.60)}{1019}}=0.630[/tex]

The 95% confidence interval for the true proportion would be given by (0.570;0.630) .

And if we convert this into % we got (57.0%, 63.0%)

Step-by-step explanation:

The information given we have the following info given:

[tex] n = 1019[/tex] represent the sampel size

[tex] \hat p=0.6[/tex] represent the sample proportion of interest

The confidence level is 95%, our significance level would be given by [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2 =0.025[/tex]. And the critical value would be given by:

[tex]z_{\alpha/2}=-1.96, z_{1-\alpha/2}=1.96[/tex]

The confidence interval for the mean is given by the following formula:  

[tex]\hat p \pm z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]

Replacing the info given we got:

[tex]0.60 - 1.96\sqrt{\frac{0.60(1-0.60)}{1019}}=0.570[/tex]

[tex]0.60 + 1.96\sqrt{\frac{0.60(1-0.60)}{1019}}=0.630[/tex]

The 95% confidence interval for the true proportion would be given by (0.570;0.630) .

And if we convert this into % we got (57.0%, 63.0%)

Express this number in scientific notation 0.0008235

Answers

8.235*10^-4

= 0.0008235

What is the mean (average) of the following data set? 3, 5, 7, 8, 9, 4, 2, 1, 3, 8 *

Answers

Answer:

5

Step-by-step explanation:

3+5+7+8+9+4+2+1+3+8=50

50/10=5

Given: m∠APC = 27°, m DC = 27° Find: m AB

Answers

Answer:

The unknown angle = 126°

Step-by-step explanation:

From angles in a triangle.

The total sum of angles in a triangle = 180°

Let the unknown angle be x

27 + 27 + x = 180

X = 180-27-27

X= 180-54

X = 126°

Answer: 81°

Step-by-step explanation:

m<p is a secant exterior angle which equals arcAB-arcDC divided by 2.

plug the numbers in and arcAB-27 / 2 =54 Arc AB is 27+54=81 degrees

Is p = –11 a solution to the inequality below?

p
> –5

Answers

Answer:

not a solution

Step-by-step explanation:

p > -5

Let p = -11

-11 > -5

This is a false statement

It is not a solution

I’ve been stuck on this for days plz help:,(

Answers

Answer:

a = 27

b = 81

Step-by-step explanation:

3^{3} = 27

3^{4} = 81

A wedge of cheese in the shape of a triangular prism is shown below.
​Find the surface area of the cheese.

_______Square feet

Answers

Answer:

[tex]144 $ Square feet[/tex]

Step-by-step explanation:

The given triangular prism comprises of two equal right triangles and three rectangles.

The Surface Area of the Prism

=Area of Two Triangles + Area of Three Rectangles

[tex]=2(0.5X6X8)+(6X4)+(10X4)+(8X4)\\=48+24+40+32\\=144 $ Square feet[/tex]

Answer: 144

Step-by-step explanation:32+24+4-+48=144

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