Chapter 7.4, Problem 50PPS

(a)

To determine

To find: The polynomial that represents the volume of the rocket.

Answer to Problem 50PPS

The polynomial that represent the volume of the rocket is $\left(t+\frac{h}{3}\right)\pi r^2$ .

Explanation of Solution

Given information:

Rocky and Arturo are designing a rocket for a competition. The volume of the cone is the product of $\frac{1}{3}$ , $\pi$ , the height $\left(h\right)$ , and the square of the radius $\left(r\right)$ The volume of cylinder is the product of $\pi$ , the height $\left(t\right)$ , and square of the radius $\left(r\right)$ .

  

Figure $\left(1\right)$

Calculation:

Consider the question

Volume of the cylindrical shape $=\pi r^{2}t$

And volume of cone shape $=\frac{1}{3}\pi r^{2}h$

The volume of the rocket $\left(R\right)$$=$ volume of the cylinder shape $+$ volume of cone shape

  $\begin{array}{l}R=\pi r^{2}t+\frac{1}{3}\pi r^{2}h\\ R=\left(t+\frac{h}{3}\right)\pi r^2\end{array}$

Therefore, the polynomial that represent the volume of the rocket is $\left(t+\frac{h}{3}\right)\pi r^2$ .

(b)

To determine

To find: The volume of the rocket.

Answer to Problem 50PPS

The volume of the rocket is $282.74$ cubic inches.

Explanation of Solution

Given information:

Rocky and Arturo are designing a rocket for a competition. The volume of the cone is the product of $\frac{1}{3}$ , $\pi$ , the height $\left(h\right)$ , and the square of the radius $\left(r\right)$ The volume of cylinder is the product of $\pi$ , the height $\left(t\right)$ , and square of the radius $\left(r\right)$ .And the height of the rocket is $8$ inches, the height of the top is $6$ inches and radius is $3$ inches.

Calculation:

The volume of the rocket is $\left(t+\frac{h}{3}\right)\pi r^2$ .

Substitute the values.

  $R=\left(8+\frac{6}{3}\right)\pi \times 3^2$

  $\begin{array}{l}R=\left(8+2\right)\pi \times 9\\ R=90\pi \\ R=282.74\end{array}$

Therefore, the volume of the rocket is $282.74$ cubic inches.

(c)

To determine

To find: The volume of the rocket.

Answer to Problem 50PPS

The volume of the rocket is $536.16$ cubic inches.

Explanation of Solution

Given information:

Rocky and Arturo are designing a rocket for a competition. The volume of the cone is the product of $\frac{1}{3}$ , $\pi$ , the height $\left(h\right)$ , and the square of the radius $\left(r\right)$ The volume of cylinder is the product of $\pi$ , the height $\left(t\right)$ , and square of the radius $\left(r\right)$ .And the height of the rocket is $9$ inches, the height of the top is $5$ inches and radius is $4$ inches.

Calculation:

The volume of the rocket is $\left(t+\frac{h}{3}\right)\pi r^2$ .

Substitute the values.

  $R=\left(9+\frac{5}{3}\right)\pi \times 4^2$

  $\begin{array}{l}R=\left(\frac{32}{3}\right)\pi \times 16\\ R=\frac{512}{3}\pi \\ R=536.16\end{array}$

Therefore, the volume of the rocket is $536.16$ cubic inches.