Question

find the volume of the object shown below, consisting of a hemisphere with a cone carved out of the top. use 3.14 for π.
30 ft
9 ft
the volume of the solid, rounded to the nearest hundred, is about ______ cubic feet.
the solution is box

Explanation:

Step1: Find radius of hemisphere and cone

The diameter of the hemisphere (and the base of the cone) is 30 ft, so the radius \( r = \frac{30}{2}=15 \) ft. The height of the cone \( h = 9 \) ft.

Step2: Volume of hemisphere

The formula for the volume of a hemisphere is \( V_{hemisphere}=\frac{2}{3}\pi r^{3} \). Substituting \( \pi = 3.14 \) and \( r = 15 \):
\[ \begin{align*} V_{hemisphere}&=\frac{2}{3}\times3.14\times15^{3}\\ &=\frac{2}{3}\times3.14\times3375\\ &=2\times3.14\times1125\\ & = 6.28\times1125\\ &=7065 \end{align*} \]

Step3: Volume of cone

The formula for the volume of a cone is \( V_{cone}=\frac{1}{3}\pi r^{2}h \). Substituting \( \pi = 3.14 \), \( r = 15 \), and \( h = 9 \):
\[ \begin{align*} V_{cone}&=\frac{1}{3}\times3.14\times15^{2}\times9\\ &=\frac{1}{3}\times3.14\times225\times9\\ &=3.14\times225\times3\\ &=3.14\times675\\ & = 2119.5 \end{align*} \]

Step4: Volume of the object

The volume of the object is the volume of the hemisphere minus the volume of the cone: \( V = V_{hemisphere}-V_{cone}=7065 - 2119.5=4945.5 \). Rounding to the nearest hundred, we look at the tens digit (4) and since it is less than 5, we round down? Wait, no, 4945.5: the hundreds place is 9, tens is 4. Wait, 4945.5 is between 4900 and 5000. The distance to 4900 is \( 4945.5 - 4900 = 45.5 \), the distance to 5000 is \( 5000 - 4945.5 = 54.5 \). Wait, no, wait, 4945.5 rounded to the nearest hundred: the number is 4945.5. The hundreds digit is 9 (in 4 9 45.5), the next digit is 4 (tens place). Since 4 < 5, we keep the hundreds digit as 9, but wait, 4945.5 is closer to 4900? Wait, no, 4945.5: 4900 is 45.5 less, 5000 is 54.5 more. Wait, but 4945.5 is 4945.5, so when rounding to the nearest hundred, we look at the tens digit (4) and the ones digit (5). Wait, the rule is: look at the digit in the tens place (for rounding to nearest hundred, we look at the digit in the tens place, which is the second digit from the right in the number before the decimal? Wait, 4945.5: the number is 4 (thousands), 9 (hundreds), 4 (tens), 5 (ones),.5 (tenths). So to round to the nearest hundred, we look at the tens digit (4). Since 4 < 5, we round down the hundreds place? Wait, no, 4945.5 is 4900 + 45.5. So rounding to the nearest hundred, it's 4900? Wait, but wait, 4945.5 is closer to 4900 than 5000? Wait, 4945.5 - 4900 = 45.5, 5000 - 4945.5 = 54.5. So yes, 4945.5 is closer to 4900? Wait, no, 45.5 < 54.5, so yes, round to 4900? Wait, but wait, maybe I made a mistake in calculation. Wait, let's recalculate the volumes.

Wait, hemisphere volume: \( \frac{2}{3}\pi r^{3} \), \( r = 15 \), so \( r^{3}=3375 \), \( \frac{2}{3}\times3375 = 2250 \), \( 2250\times3.14 = 7065 \). Correct. Cone volume: \( \frac{1}{3}\pi r^{2}h \), \( r = 15 \), \( r^{2}=225 \), \( h = 9 \), \( \frac{1}{3}\times225\times9 = 225\times3 = 675 \), \( 675\times3.14 = 2119.5 \). Correct. Then 7065 - 2119.5 = 4945.5. Now, 4945.5 rounded to the nearest hundred: the hundreds digit is 9 (in 4 9 45.5), the tens digit is 4. Since 4 < 5, we round down the hundreds place? Wait, no, when rounding to the nearest hundred, we look at the digit in the tens place (4) to decide. If the tens digit is less than 5, we keep the hundreds digit as it is and set the tens and ones digits to zero. Wait, but 4945.5: the number is 4945.5, so to the nearest hundred, it's 4900? Wait, but 4945.5 is 4945.5, which is 49 hundred and 45.5. So 4945.5 is closer to 4900 than 5000? Wait, 4945.5 - 4900 = 45.5, 5000 - 4945.5 = 54.5. So yes, 4945.5 rounds to 4900? Wait, but maybe the problem expects rounding to the nearest hundred, so 4945.5 is approximately 4900? Wait, no, wait, 4945.5 is 4945.5, so when rounding to the nearest hundred, we look at the digit in the tens place (4) and the digit in the ones place (5). Wait, the rule is: if the digit in the tens place is 5 or more, we round up the hundreds place. Wait, no, the tens digit is 4, which is less than 5, so we round down. So 4945.5 rounded to the nearest hundred is 4900? Wait, but let's check again. Wait, 4945.5: the hundreds digit is 9 (position 3 from the right: 5 (ones), 4 (tens), 9 (hundreds), 4 (thousands)). So to round to the nearest hundred, we look at the tens digit (4). Since 4 < 5, we keep the hundreds digit as 9 and set the tens and ones digits to 0. So 4900. But wait, 4945.5 is 4945.5, which is 4945.5, so 4945.5 is 4900 + 45.5, so yes, rounds to 4900. Wait, but maybe I made a mistake in the problem. Wait, the hemisphere has diameter 30, so radius 15, correct. The cone has height 9, radius 15, correct. So the calculations seem right.

Answer:

4900