Question:
A car is being driven, in a straight line and at a uniform speed, towards the base of a vertical tower. The top of the tower is observed from the car and, in the process, it takes 10 min for the angle of elevation to change from 45\( ^\circ \) to 60\( ^\circ \). After how much more time will this car reach the base of the tower?
A car is being driven, in a straight line and at a uniform speed, towards the base of a vertical tower. The top of the tower is observed from the car and, in the process, it takes 10 min for the angle of elevation to change from 45\( ^\circ \) to 60\( ^\circ \). After how much more time will this car reach the base of the tower?
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Let the tower height be h. The car is at distance h from the base when the angle is 45 degrees and at distance h divided by root 3 when it is 60 degrees.
Updated On: Nov 6, 2026
- \( 6(\sqrt{3}+\sqrt{2}) \)
- \( 5(\sqrt{3}+1) \)
- \( 7(\sqrt{3}-1) \)
- \( 8(\sqrt{3}-2) \)
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The Correct Option is B
Solution and Explanation
Step 1: Distances from the base:
Let the height of the tower be h. At \( 45^\circ \): distance \( d_1 = h/\tan 45^\circ = h \). At \( 60^\circ \): distance \( d_2 = h/\tan 60^\circ = h/\sqrt{3} \).
Step 2: Speed:
In 10 min the car covers \( d_1 - d_2 = h\left(1 - \dfrac{1}{\sqrt{3}}\right) = h\,\dfrac{\sqrt{3}-1}{\sqrt{3}} \).
Step 3: Remaining time:
Remaining distance is \( d_2 = h/\sqrt{3} \). Time \( = 10 \times \dfrac{h/\sqrt{3}}{h(\sqrt{3}-1)/\sqrt{3}} = \dfrac{10}{\sqrt{3}-1} \).
Step 4: Simplify:
\( \dfrac{10}{\sqrt{3}-1} \times \dfrac{\sqrt{3}+1}{\sqrt{3}+1} = \dfrac{10(\sqrt{3}+1)}{2} = 5(\sqrt{3}+1) \) minutes (about 13.66 min).
Final Answer:
The correct option is (2): \( 5(\sqrt{3}+1) \)
Let the height of the tower be h. At \( 45^\circ \): distance \( d_1 = h/\tan 45^\circ = h \). At \( 60^\circ \): distance \( d_2 = h/\tan 60^\circ = h/\sqrt{3} \).
Step 2: Speed:
In 10 min the car covers \( d_1 - d_2 = h\left(1 - \dfrac{1}{\sqrt{3}}\right) = h\,\dfrac{\sqrt{3}-1}{\sqrt{3}} \).
Step 3: Remaining time:
Remaining distance is \( d_2 = h/\sqrt{3} \). Time \( = 10 \times \dfrac{h/\sqrt{3}}{h(\sqrt{3}-1)/\sqrt{3}} = \dfrac{10}{\sqrt{3}-1} \).
Step 4: Simplify:
\( \dfrac{10}{\sqrt{3}-1} \times \dfrac{\sqrt{3}+1}{\sqrt{3}+1} = \dfrac{10(\sqrt{3}+1)}{2} = 5(\sqrt{3}+1) \) minutes (about 13.66 min).
Final Answer:
The correct option is (2): \( 5(\sqrt{3}+1) \)
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