Question

if the radius of a cone is 5 cm and its height is 10 cm, what is the volume of the cone?
a. 50πcm³
b. $\frac{250}{3}$πcm³
c. 100πcm³
d. $\frac{100}{3}$πcm³
which principle states that two solids with the same height and identical cross - sectional areas at every level have the same volume?
a. cavalieris principle
b. pythagorean theorem
c. volume theorem
d. area congruence principle
if the volume of a cone is 36πcm³ and its height is 6 cm, what is its radius? (use π ≈ 3.14)
a. 6 cm
b. 3 cm
c. 4.24 cm
d. 5.20 cm
which of the following statements is true about cavalieris principle?
a. if two solids have the same height and every cross - sectional area at the same height is equal, then the two solids have the same volume.
b. if two solids have the same height and every cross - sectional area at the same height is equal, then the two solids have different volumes.
c. if two solids have different heights and different cross - sectional areas at the same height, they have the same volume.
d. if two solids have different heights and equal cross - sectional areas at the same height, they have different volumes.

Explanation:

Step1: Recall cone volume formula

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

Step2: Substitute $r=5, h=10$

$V=\frac{1}{3}\pi (5)^2 (10)=\frac{1}{3}\pi \times25\times10=\frac{250}{3}\pi$
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Step1: Identify the volume principle

Match the definition to the principle name.
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Step1: Rearrange cone volume formula

Solve $V=\frac{1}{3}\pi r^2 h$ for $r$: $r=\sqrt{\frac{3V}{\pi h}}$

Step2: Substitute $V=36\pi, h=6$

$r=\sqrt{\frac{3\times36\pi}{\pi \times6}}=\sqrt{\frac{108}{6}}=\sqrt{18}=3\sqrt{2}\approx4.24$
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Step1: Verify Cavalieri's Principle

Check which option matches the principle's definition.

Answer:

1. b. $\frac{250}{3}\pi cm^3$
2. a. Cavalieri's Principle
3. c. 4.24 cm
4. a. If two solids have the same height and every cross-sectional area at the same height is equal, then the two solids have the same volume.