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A Hydrogenator water gun has a cylindrical water tank, which is 30 centimeters long. Using a hose, Jack fills his Hydrogenator with π cubic centimeters of his water tank every second. If it takes him 8 minutes to fill the tank with water, what is the diameter of the circular base of the gun’s water tank?
A.4 cm
B.10 cm
C.8 cm
D.6 cm
Answer
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Hint:In the above question we have a water tank in a cylindrical shape whose length is given. We will assume the radius of this water tank. Then we will put the value of length and radius the formula of Volume of cylinder which is \[\pi {r^2}h\]( in this way we can form the equation 1).Now we have the rate of filling the tank and time taken to fill it, so we can find the volume easily by multiplying the above two values that is rate of filling and time taken. This would be our second equation. At last we will solve these two equations to find the value of diameter.
Complete step-by-step answer:
Given that,
Length of cylindrical water tank = 30 cm
Let its radius be r
Volume of cylindrical water tank \[ = \pi {r^2}h\]
$ = \left( {\pi {r^2} \times 30} \right)$cubic cm ….(i)
Jack fills his Hydrogenator at the rate of = π cubic cm per second
Total time taken to fill the tank = 8 min = 8 X 60 = 480 seconds
Total volume filled = (rate of filling X Time taken to fill the tank)
Thus, total volume filled $ = \left( {\pi \times 480} \right)$ cubic cm …(ii)
From equations (i) and (ii), we get;
$\pi {r^2} \times 30 = \pi \times 480$
$ \Rightarrow {r^2} = \dfrac{{480}}{{30}} = 16$cm
$ \Rightarrow r = 4$cm
Radius = 4 cm
Hence, the diameter of the circular base of the gun’s water tank is the twice of the radius $ = 2 \times 4 = 8$cm
So, the correct answer is “Option C”.
Note:In these types of questions, we should equate volume as volume of water remains the same. Also remember, the volume of a Cylinder is $\pi r^2 h$ , where r is the radius of the base and h is the height or length of a cylinder.
Complete step-by-step answer:
Given that,
Length of cylindrical water tank = 30 cm
Let its radius be r
Volume of cylindrical water tank \[ = \pi {r^2}h\]
$ = \left( {\pi {r^2} \times 30} \right)$cubic cm ….(i)
Jack fills his Hydrogenator at the rate of = π cubic cm per second
Total time taken to fill the tank = 8 min = 8 X 60 = 480 seconds
Total volume filled = (rate of filling X Time taken to fill the tank)
Thus, total volume filled $ = \left( {\pi \times 480} \right)$ cubic cm …(ii)
From equations (i) and (ii), we get;
$\pi {r^2} \times 30 = \pi \times 480$
$ \Rightarrow {r^2} = \dfrac{{480}}{{30}} = 16$cm
$ \Rightarrow r = 4$cm
Radius = 4 cm
Hence, the diameter of the circular base of the gun’s water tank is the twice of the radius $ = 2 \times 4 = 8$cm
So, the correct answer is “Option C”.
Note:In these types of questions, we should equate volume as volume of water remains the same. Also remember, the volume of a Cylinder is $\pi r^2 h$ , where r is the radius of the base and h is the height or length of a cylinder.
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