Question

the volume of an object is equal to the ratio of its mass to density, $v = \frac{m}{d}$. the mass of a spherical grape is 8.4 grams and its density is 2 grams per cubic centimeter. what is the radius of the grape? round to the nearest tenth of a centimeter. \bigcirc 1.0 cm \bigcirc 1.5 cm \bigcirc 1.9 cm \bigcirc 2.1 cm

Explanation:

Step1: Calculate the volume of the grape

Given the formula \( V=\frac{m}{d} \), where \( m = 8.4 \) grams and \( d=2 \) grams per cubic centimeter. Substitute the values into the formula:
\( V=\frac{8.4}{2}=4.2 \) cubic centimeters.

Step2: Use the volume formula for a sphere to find the radius

The volume formula for a sphere is \( V = \frac{4}{3}\pi r^{3} \). We know \( V = 4.2 \), so we can solve for \( r \).
First, rearrange the formula to solve for \( r^{3} \):
\( r^{3}=\frac{3V}{4\pi} \)
Substitute \( V = 4.2 \) into the formula:
\( r^{3}=\frac{3\times4.2}{4\pi}=\frac{12.6}{4\pi}\approx\frac{12.6}{12.566}\approx1.003 \)
Then, take the cube root of both sides to find \( r \):
\( r=\sqrt[3]{1.003}\approx1.0 \)? Wait, no, wait, maybe I made a mistake. Wait, let's recalculate \( \frac{3\times4.2}{4\pi} \). \( 3\times4.2 = 12.6 \), \( 4\pi\approx12.566 \), so \( \frac{12.6}{12.566}\approx1.0027 \). Wait, that can't be right. Wait, maybe I messed up the volume. Wait, no, the mass is 8.4, density 2, so volume is 4.2. Wait, but let's check the sphere volume formula again. Wait, maybe I miscalculated. Wait, no, let's do it again.

Wait, \( V=\frac{4}{3}\pi r^{3} \), so \( r^{3}=\frac{3V}{4\pi} \). So \( V = 4.2 \), so \( 3V = 12.6 \), \( 4\pi\approx12.566 \), so \( r^{3}=\frac{12.6}{12.566}\approx1.0027 \), so \( r\approx\sqrt[3]{1.0027}\approx1.0 \)? But that's one of the options, but let's check again. Wait, maybe I made a mistake in the volume. Wait, mass is 8.4, density 2, so volume is 8.4/2 = 4.2. That's correct. Then sphere volume is 4/3 π r³ = 4.2. So 4/3 π r³ = 4.2. Multiply both sides by 3: 4π r³ = 12.6. Then divide by 4π: r³ = 12.6/(4π) ≈ 12.6/12.566 ≈ 1.0027. Then cube root of 1.0027 is approximately 1.0. But wait, the options have 1.0, 1.5, 1.9, 2.1. Wait, maybe I made a mistake in the volume calculation. Wait, no, mass is 8.4, density 2, so volume is 4.2. Wait, maybe the grape is not a perfect sphere? No, the problem says spherical grape. Wait, maybe I miscalculated the cube root. Wait, let's check with r = 1.0: volume is 4/3 π (1)³ ≈ 4.188, which is close to 4.2. So r ≈ 1.0. But wait, the options have 1.0 as an option. Wait, but let's check again. Wait, 8.4 divided by 2 is 4.2. Then 4/3 π r³ = 4.2. So r³ = (4.2 * 3)/(4π) = 12.6/(12.566) ≈ 1.0027. So r ≈ 1.0. So the answer should be 1.0 cm.

Wait, but maybe I made a mistake. Let's check with r = 1.5: volume is 4/3 π (1.5)³ = 4/3 π (3.375) = 4.5 π ≈ 14.14, which is way bigger than 4.2. So r = 1.0 is correct.

Answer:

1.0 cm