Question

geometry question 34, 6.5.5 part 2 of 2 hw score: 74.14%, 43 of 58 points points: 0 of 1 save find the volume of the solid. for formulas containing π, give the exact answer and then an approximation using \\(\frac{22}{7}\\) for π. image of a cone with height 9 yd and radius 2 yd the exact value of the volume is 12π cubic yards. (simplify your answer. type an exact answer in terms of π.) the approximate value of the volume is \\(\square\\) (simplify your answer. type a whole number, fraction, or mixed number.)

Explanation:

Step1: Recall the volume formula for a cone

The volume \( V \) of a cone is given by \( V=\frac{1}{3}\pi r^{2}h \). We know the exact volume is \( 12\pi \), and we need to approximate it using \( \pi=\frac{22}{7} \). So we substitute \( \pi \) in the exact volume formula.
The exact volume \( V = 12\pi \), substitute \( \pi=\frac{22}{7} \), we get \( V=12\times\frac{22}{7} \).

Step2: Calculate the product

First, calculate \( 12\times22 = 264 \), then divide by 7: \( \frac{264}{7}=37\frac{5}{7}\approx37.71 \), but wait, wait, maybe I made a mistake. Wait, the formula for the volume of a cone is \( V = \frac{1}{3}\pi r^{2}h \). Let's check the radius and height. From the diagram, radius \( r = 2 \) yd, height \( h=9 \) yd. Then \( V=\frac{1}{3}\pi(2)^{2}(9)=\frac{1}{3}\pi\times4\times9 = 12\pi \), that's correct. Now to approximate, substitute \( \pi=\frac{22}{7} \), so \( V = 12\times\frac{22}{7}=\frac{264}{7}=37\frac{5}{7}\approx37.71 \), but as a fraction, \( \frac{264}{7}=37\frac{5}{7} \), but maybe the exact volume was miscalculated? Wait no, the exact volume is given as \( 12\pi \), so we just need to compute \( 12\times\frac{22}{7} \).

Wait, \( 12\times\frac{22}{7}=\frac{264}{7}=37\frac{5}{7} \), but let's do the multiplication again. \( 12\times22 = 264 \), divided by 7 is \( 37\frac{5}{7} \), which is approximately 37.71, but as a fraction, it's \( \frac{264}{7} \) or \( 37\frac{5}{7} \). But maybe the problem expects a whole number? Wait no, the problem says "Type a whole number, fraction, or mixed number." So \( 12\times\frac{22}{7}=\frac{264}{7}=37\frac{5}{7} \), but let's check again. Wait, the radius is 2, height is 9. Volume of cone is \( \frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\times4\times9 = 12\pi \), correct. Then approximate with \( \pi=\frac{22}{7} \), so \( 12\times\frac{22}{7}=\frac{264}{7}=37\frac{5}{7} \approx 37.71 \), but as a fraction, it's \( \frac{264}{7} \) or \( 37\frac{5}{7} \).

Wait, maybe I made a mistake in the radius? Wait the diagram shows radius 2 yd, height 9 yd. So yes, \( r = 2 \), \( h = 9 \). So \( V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\times4\times9 = 12\pi \). Then approximate: \( 12\times\frac{22}{7}=\frac{264}{7}=37\frac{5}{7} \). So the approximate value is \( \frac{264}{7} \) or \( 37\frac{5}{7} \).

Answer:

\( \frac{264}{7} \) (or \( 37\frac{5}{7} \))