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cupoosta [38]
3 years ago
11

Which graph shows the volumes, V, of rectangular prisms with lengths of 2 centimeters, widths of 3 centimeters, and heights of h

?
Mathematics
1 answer:
Ksivusya [100]3 years ago
7 0

Answer:

Its C

Step-by-step explanation:

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Read 2 more answers
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
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