-h (6h-1)<br> please hurry I really need help!

If you are offered one slice from a round pizza (in other words, a sector of a circle) and the slice must have a perimeter of 24

VARVARA [1.3K]

Answer:

A 12 in diameter will reward you with the largest slice of pizza.

Step-by-step explanation:

Let $r$ be the radius and $\theta$ be the angle of a circle.

According with the graph, the area of the sector is given by

$A=\frac{1}{2}r^2\theta$

The arc lenght of a circle with radius $r$ and angle $\theta$ is $r \theta$

The perimeter of the pizza slice is composed of two straight pieces, each of length r inches, and an arc of the circle which you know has length s = rθ inches. Thus the perimeter has length

$2r+r\theta=24 \:in$

We need to express the area as a function of one variable, to do this we use the above equation and we solve for $\theta$

$2r+r\theta=24\\r\theta=24-2r\\\theta=\frac{24-2r}{r}$

Next, we substitute this equation into the area equation

$A=\frac{1}{2}r^2(\frac{24-2r}{r})\\A=\frac{1}{2}r(24-2r)\\A=12r-r^2$

The domain of the area is

$0$

To find the diameter of pizza that will reward you with the largest slice you need to find the derivative of the area and set it equal to zero to find the critical points.

$\frac{d}{dr} A=\frac{d}{dr}(12r-r^2)\\A'(r)=\frac{d}{dr}(12r)-\frac{d}{dr}(r^2)\\A'(r)=12-2r$

$12-2r=0\\-2r=-12\\\frac{-2r}{-2}=\frac{-12}{-2}\\r=6$

To check if $r=6$ is a maximum we use the Second Derivative test

if $f'(c)=0$ and $f''(c)$ , then f(x) has a local maximum at x = c.

The second derivative is

$\frac{d}{dr} A'(r)=\frac{d}{dr} (12-2r)\\A''(r)=-2$

Because $A''(r)=-2$ the largest slice is when r = 6 in.

The diameter of the pizza is given by

$D=2r=2\cdot 6=12 \:in$

A 12 in diameter will reward you with the largest slice of pizza.

3 0

3 years ago

Find the area of the figure.

PtichkaEL [24]

Answer:

which figure?........

7 0

3 years ago

How many pieces 3 and three fifths in. long can be cut from 28 metal rods each 15 in. long? Disregard waste.

Murrr4er [49]

Answer: 112 pieces

Step-by-step explanation:

1. Rewrite $3^{\frac{3}{5}}$ as a decimal. Divide the numerator and the denominator of the fractional part and add it to the integer part:

$3+0.6=3.6in$

2. You know that each metal rod is 15 inches long. Then, you must divide the lenght of each one by 3.6:

$pieces=\frac{15in}{3.6in}4.16$

3. The waste is 0.16, therefore, you obtain 4 pieces from a metal rod. The total pieces obtained from 28 metal rods are:

$total=4pieces*28=112pieces$

8 0

3 years ago

Read 2 more answers

X − 78 = −42<br>x = 78 − 42<br>x = −36

WITCHER [35]

Answer:

x= 36

(it is positive since 78 is larger than 42, everything else is correct)

Step-by-step explanation:

5 0

3 years ago

What is the solution to the system of equations 2x-y=7 and y=2x+3

Yuri [45]

Make y the subject in both equations

$2x-y=7$

$-y=-2x+7$

$y=2x-7$

For the second equation

$y=2x+3$

The two equations are parallel because they have the same gradient or slope but different y intercepts.

This type of equations have no solution because they will never intersect.

$NO SOLUTION$

The correct answer is C

6 0

3 years ago

-h (6h-1) please hurry I really need help!