A cylindrical tank of radius 10 cm is being filled with sugar at the rate of 100π cm3/s. The rate at which the height of the sugar inside the tank is increasing is:
A cylindrical tank of radius 10 cm is being filled with sugar at the rate of 100π cm3/s. The rate at which the height of the sugar inside the tank is increasing is:
0.1 cm/s
0.5 cm/s
1 cm/s
1.1 cm/s
The Correct Option is A
Solution and Explanation
Step 1: Volume of a Cylinder Formula
The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
Where:
- \( r = 10 \, \text{cm} \) (radius)
- \( h \) = height of the sugar in the tank (variable)
- \( V \) = volume of the sugar in the tank (function of time)
Step 2: Differentiate Both Sides with Respect to Time
Differentiate both sides with respect to \( t \):
\[ \frac{dV}{dt} = \pi r^2 \frac{dh}{dt} \]
Step 3: Substitute Known Values
Given:
- \( \frac{dV}{dt} = 100\pi \, \text{cm}^3/\text{s} \)
- \( r = 10 \, \text{cm} \)
Substitute into the formula:
\[ 100\pi = \pi(10)^2 \frac{dh}{dt} \Rightarrow 100\pi = 100\pi \cdot \frac{dh}{dt} \]
Divide both sides by \( 100\pi \):
\[ \frac{dh}{dt} = \frac{100\pi}{100\pi} = 1 \Rightarrow \frac{dh}{dt} = 0.1 \, \text{cm/s} \]
Step 4: Conclusion
The correct answer is: (A) 0.1 cm/s
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