The cost of the fuel for running the engine of an army tank is proport

Question

The cost of the fuel for running the engine of an army tank is proportional to the square of the speed and is equal to $64 per hour at a speed of 16 kmph. The total cost for the tank per hour equals to the cost of the fuel per hour plus $400 per hour. The tank has to make a journey of 400 km at a constant speed. What is the most cost-efficient speed for this journey ?

A. 10 kmph

B. 20 kmph

C. 35 kmph

D. 40 kmph

E. 50 kmph

A. 10 kmph

B. 20 kmph

C. 35 kmph

D. 40 kmph

E. 50 kmph

Abhishek009 · Answer

C = ks^2

Or,  64 = k*16*16

Or,  k = \frac{1}{4}

C = ks^2

Or,  400 = \frac{1}{4}s^2

So,  s = 40 , Answer must be (D)

UsamaGuddar · Answer

Abhishek009 , where do you get C=400? Can you pls explain

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Abhishek009 · Answer

I took cost as $400

ScottTargetTestPrep · Answer

Solution:

Let’s let c = the cost of the fuel per hour and s = the speed of the tank. We are told that cost is proportional to the square of the speed, so we have:

c = k x s^2,

where k = the constant of proportionality.

Substituting the given information, we have:

64 = k x 16^2

64/256 = k

1/4 = k

We must calculate the per-hour cost, the length of the trip (in hours), and the total cost for the trip for each of the answer choices.

Choice A: 10 kmph: c = ¼ x 10^2 = $25 per hour fuel cost + $400 = $425 per hour.
Length of trip: 400/10 = 40 hours, so the total trip cost is 425 x 40 = $17,000.

Choice B: 20 kmph: c = ¼ x 20^2 = $100 per hour fuel cost + $400 = $500 per hour.
Length of trip: 400/20 = 20 hours, so the total trip cost is 500 x 20 = $10,000.

Choice C: 35 kmph: c = ¼ x 35^2 = $306.25 per hour fuel cost + $400 = $706.25 per hour.
Length of trip: 400/35 = 11.43 hours, so the total trip cost is 706.25 x 11.43 ≈ $8,072.

Choice D: 40 kmph: c = ¼ x 40^2 = $400 per hour fuel cost + $400 = $800 per hour.
Length of trip: 400/40 = 10 hours, so the total trip cost is 800 x 10 = $8,000.

Choice E: 50 kmph: c = ¼ x 50^2 = $625 per hour fuel cost + $400 = $1025 per hour.
Length of trip: 400/50 = 8 hours, so the total trip cost is 1025 x 8 = $8,200.

We see that a speed of 40 kmph is the most cost-efficient speed of the given choices.

Answer: D

CharuGMATKING · Answer

An alternative way if you do not want to substitute the choices, after some calculation you will find that you need to minimize the value of (x^2 +1600)/x. Differentiating will give you that the value for x = 40.
(Because the differential will be 1 - 1600/x^2 = 0)
Again this is an extreme step that ideally you need not do, just wanted to give another way.

hsr071 · Answer

proportionality const = 1/4

Fuel Cost per hour:
C = 1⁄4*s2

dist = 400 => s×t=400 => t=400/s

Tank Charges = t×400 =400^2/s

Journey Cost
Cj = Fuel Cost per hour × time + Tank Charge
Cj = s2/4×t + 400^2/s = s2/4×400/s + 400^2/s = 100s + 400^2/s

For min Cj, dCj/ds = 0
100 - 400^2/s2=0
100 = 400^2/s2
s2 = 400×400/100 = 400×4 = 20×2 = 40

The cost of the fuel for running the engine of an army tank is proport

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The cost of the fuel for running the engine of an army tank is proport

Prep Toolkit

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