Question

If a cone of height h and a sphere have same radii r and same volume, then r:h will be:

• A. 4:1
• B. 1:4
• C. 2:3
• D. 3:2

Hint:

This problem is a direct application of volume formulas. Always write down the formulas, set up the equality as per the problem statement, and then simplify carefully. Remember that $r^3 = r^2 \times r$.

Answer 1

The Correct Option is B

Step 1: Write down the formulas for the volume of a cone and a sphere.
Volume of a cone (\( V_{\text{cone}} \)) = \(\frac{1}{3} \pi r^2 h\) Volume of a sphere (\( V_{\text{sphere}} \)) = \(\frac{4}{3} \pi r^3\) where \( r \) is the radius and \( h \) is the height. Step 2: Use the given condition that their volumes are the same.
Given that \( V_{\text{cone}} = V_{\text{sphere}} \): \[ \frac{1}{3} \pi r^2 h = \frac{4}{3} \pi r^3 \] Step 3: Simplify the equation to find the ratio \( r : h \).
First, cancel common terms on both sides. Both sides have \(\frac{1}{3}\) and \(\pi\). Also, both sides have \( r^2 \): \[ r^2 h = 4 r^3 \] Divide both sides by \( r^2 \) (assuming \( r \neq 0 \)): \[ h = 4r \] Now, find the ratio \( r : h \). Divide both sides by \( h \): \[ 1 = 4 \frac{r}{h} \] Divide both sides by 4: \[ \frac{1}{4} = \frac{r}{h} \] So, the ratio \( r : h \) is \( 1 : 4 \). Step 4: Final Answer. The ratio \( r : h \) is \( 1 : 4 \). \[ (2) 1:4 \]