Question

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 4 yd the exact value of the volume is 48π cubic yards. (simplify your answer. type an exact answer in terms of π.) the approximate value of the volume is \\(\square\\) dropdown with options: yards, square yards, cubic yards. (simplify your answer. type a whole number.)

Explanation:

Step1: Substitute π with 22/7

We know the exact volume is \( 48\pi \) cubic yards. Now we substitute \( \pi = \frac{22}{7} \) into the formula. So the volume becomes \( 48\times\frac{22}{7} \).

Step2: Calculate the product

First, multiply 48 and 22: \( 48\times22 = 1056 \). Then divide by 7: \( \frac{1056}{7} \approx 150.857 \). But we need to round to a whole number, so \( \frac{1056}{7}=150\frac{6}{7}\approx151 \)? Wait, no, wait. Wait, 48 times 22 is 1056, 1056 divided by 7: 7150=1050, 1056-1050=6, so 150 + 6/7 ≈150.857, but wait, maybe I made a mistake. Wait, the formula for the volume of a cone is \( \frac{1}{3}\pi r^2 h \). Let's check the exact volume again. The radius is 4 yd, height is 9 yd. So \( V=\frac{1}{3}\pi(4)^2(9)=\frac{1}{3}\pi\times16\times9 = 48\pi \), that's correct. Then substituting \( \pi=\frac{22}{7} \), we have \( 48\times\frac{22}{7}=\frac{1056}{7}\approx150.857 \), which rounds to 151? Wait, no, wait, 22/7 is approximately 3.142857. 483.142857≈483 + 480.142857=144 + 6.857136=150.857136, so the whole number approximation is 151? Wait, but maybe the problem expects using 22/7 directly. Let's do 48(22/7)= (4822)/7=1056/7=150.857..., so the whole number is 151? Wait, no, maybe I miscalculated. Wait, 7150=1050, 1056-1050=6, so 150 and 6/7, which is approximately 150.86, so the whole number is 151? Wait, but let's check again. Wait, the exact volume is 48π, so with π=22/7, 48(22/7)= (48/7)22. 48 divided by 7 is approximately 6.857, times 22 is 6.85722=150.854, so yes, approximately 151. But wait, maybe the problem has a typo? Wait, no, let's check the cone volume formula again. Radius 4, height 9. \( V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi(16)(9) = 48\pi \), correct. Then 48(22/7)= (4822)/7=1056/7=150.857..., so the approximate value is 151 cubic yards. Wait, but maybe I made a mistake in the multiplication. 4822: 4022=880, 822=176, 880+176=1056, correct. 1056 divided by 7: 7150=1050, 1056-1050=6, so 150 6/7, which is approximately 150.86, so the whole number is 151.

Answer:

151 cubic yards