Question

1. if the diameter of a spherical tank is doubled, how does the volume change?

a. the volume is halved
b. the volume stays the same
c. the volume is doubled
d. the volume increases by a factor of 8

1. in the formula for the volume of a sphere, \\(\frac{4}{3}\pi r^3\\), what does r represent?

a. surface area
b. circumference
c. radius
d. diameter

1. if a sphere has a radius of 7 meters, what is its diameter?

a. 21 meters
b. 28 meters
c. 7 meters
d. 14 meters

1. if the volume of a sphere is 523.6 cubic meters, which of the following must be its radius?

a. 7 meters
b. 10 meters
c. 3 meters
d. 5 meters

1. how does the volume of a sphere relate to its surface area?

a. volume is always greater than surface area
b. surface area is calculated using volume
c. they are unrelated
d. both depend on the radius

1. which of the following units is appropriate for measuring the volume of a sphere?

a. meters
b. square centimeters
c. cubic centimeters
d. centimeters

1. what happens to the volume of a sphere if the radius is increased?

a. the volume is halved

Explanation:

Step1: Solve volume change question

Let original diameter $d$, radius $r=\frac{d}{2}$. Volume $V_1=\frac{4}{3}\pi r^3=\frac{4}{3}\pi (\frac{d}{2})^3$. New diameter $2d$, new radius $r'=d$. New volume $V_2=\frac{4}{3}\pi d^3$. Ratio $\frac{V_2}{V_1}=\frac{\frac{4}{3}\pi d^3}{\frac{4}{3}\pi (\frac{d}{2})^3}=8$.

Step2: Identify variable in sphere formula

Recall sphere volume formula definition: $r$ = radius.

Step3: Calculate diameter from radius

Diameter $=2\times$ radius. $2\times7=14$ meters.

Step4: Find radius from volume

Use $V=\frac{4}{3}\pi r^3$. Substitute $V=523.6$:
$523.6=\frac{4}{3}\times3.14\times r^3$
$r^3=\frac{523.6\times3}{4\times3.14}=125$
$r=\sqrt[3]{125}=5$ meters.

Step5: Relate sphere volume and surface area

Volume $V=\frac{4}{3}\pi r^3$, surface area $SA=4\pi r^2$; both depend on $r$.

Step6: Select volume unit

Volume uses cubic units; cubic centimeters is correct.

Answer:

1. d. The volume increases by a factor of 8
2. c. Radius
3. d. 14 meters
4. d. 5 meters
5. d. Both depend on the radius
6. c. Cubic centimeters