Question

question 16
what is the volume of a cone with a radius of 4 and a height of 15?
a. $251.33\\ \text{units}^3$
b. $39.79\\ \text{units}^3$
c. $900\\ \text{units}^3$
d. $753.98\\ \text{units}^3$

question 17
tiana would like to buy ice cream cones for her birthday party, but she wants cones with the largest volume. which of the following brands of ice cream cones should she buy?
brand a:
height of cone = 16 cm
radius of cone base = 5 cm
brand b:
height of cone = 20 cm
radius of cone base = 4 cm
a. brand a
b. brand b
c. either — the cones have equal volumes

question 18
which of the following is a solid bounded by the set of all points at a given distance from a given point?
a. cube
b. sphere
c. cylinder
d. cone

question 19
what is the surface area of a sphere with radius 2?
a. $4\pi\\ \text{units}^2$
b. $2\pi\\ \text{units}^2$
c. $8\pi\\ \text{units}^2$
d. $16\pi\\ \text{units}^2$

Explanation:

Question 16

Step1: Recall cone volume formula

The volume of a cone is $V=\frac{1}{3}\pi r^2 h$, where $r=4$, $h=15$.

Step2: Substitute values and calculate

$$\begin{align*} V&=\frac{1}{3}\pi (4)^2 (15)\\ &=\frac{1}{3}\pi \times 16 \times 15\\ &=80\pi\\ &\approx 251.33 \end{align*}$$

Question 17

Step1: Calculate Brand A volume

Use $V=\frac{1}{3}\pi r^2 h$, $r=5$, $h=16$:

$$\begin{align*} V_A&=\frac{1}{3}\pi (5)^2 (16)\\ &=\frac{400}{3}\pi\\ &\approx 418.88 \end{align*}$$

Step2: Calculate Brand B volume

Use $V=\frac{1}{3}\pi r^2 h$, $r=4$, $h=20$:

$$\begin{align*} V_B&=\frac{1}{3}\pi (4)^2 (20)\\ &=\frac{320}{3}\pi\\ &\approx 335.10 \end{align*}$$

Step3: Compare the two volumes

$V_A > V_B$, so Brand A is larger.

Question 18

Step1: Match definition to shape

A solid where all points are equidistant from a single given point is the definition of a sphere.

Question 19

Step1: Recall sphere surface area formula

The surface area of a sphere is $SA=4\pi r^2$, where $r=2$.

Step2: Substitute value and calculate

$$\begin{align*} SA&=4\pi (2)^2\\ &=4\pi \times 4\\ &=16\pi \end{align*}$$

Answer:

1. Question 16: A. 251.33 units³
2. Question 17: A. Brand A
3. Question 18: B. Sphere
4. Question 19: D. $16\pi$ units²