Question

A rectangular box of width(a), length(b), and height(c) has a solid cylinder of height 'c' and of diameter 'a'placed within it. If a= 6, b =8 and c =10,how much volume is left in the rectangular box ?

• A. 380 − 90\(\pi\)
• B. 240 − 96\(\pi\)
• C. 480 − 90\(\pi\)
• D. 480 − 180\(\pi\)

Answer 1

The Correct Option is C

To determine how much volume is left in the rectangular box after placing a solid cylinder inside it, we need to follow these steps:

1. First, calculate the volume of the rectangular box. The volume \((V_{\text{box}})\) is given by the formula: \(V_{\text{box}} = a \times b \times c\), where \(a = 6\), \(b = 8\), and \(c = 10\).
   • \(V_{\text{box}} = 6 \times 8 \times 10 = 480 \, \text{cubic units}\)
2. Next, calculate the volume of the cylinder. The volume \((V_{\text{cylinder}})\) is given by the formula: \(V_{\text{cylinder}} = \pi r^2 h\), where \(r\) is the radius of the base of the cylinder, and \(h = c\) is the height.
   • The diameter of the cylinder is \(a = 6\), so the radius \((r)\) is \(r = \frac{a}{2} = \frac{6}{2} = 3\).
   • \(V_{\text{cylinder}} = \pi \times (3)^2 \times 10 = 90\pi \, \text{cubic units}\)
3. Subtract the volume of the cylinder from the volume of the box to find the remaining volume:
   • \(V_{\text{remaining}} = V_{\text{box}} - V_{\text{cylinder}} = 480 - 90\pi \, \text{cubic units}\) 

Thus, the volume left in the rectangular box is \(480 - 90\pi\) cubic units.

The correct option is: 480 − 90\(\pi\)