Integrating both sides, we get
When
, we have
, so that
When
, we have
, which means
A = 2 a² + 2 a b
2 a² + 2 a b = 147/2
a² + a b = 36.75
a b = 36.75 - a²
b = ( 36.75 - a² ) / a
V = a² * b
V = a² * ( 36.75 - a² ) / a
V = 36.75 a - a³ ( maximum of the volume is when V` = 0 )
V` = 36.75 - 3 a²
36.75 - 3 a² = 0
3 a² = 36.75
a² = 36.75 : 3
a² = 12.25
a =√12.25
a = 3.5 ft
b = 24.5 / 3.5 = 7 ft.
Answer: The depth of the shelter for which the space inside is maximized is 3.5 ft.
This is an easy question. All required information's are provided in the question. In the first case it is given that the tank is getting filled at a rate of 3/4 gallon per 2/3 minute. So by getting the speed of filling we can compare this withe the speed at which the second tank is getting filled. The second tank fills at a rate of 5/8 gallon in 1/2 minute.
If we take the case of the first tank, then,
Speed of filling the first tank = (3/4) * (3/2)
= 9/8
So the first tank is filled at the rate of 9/8 gallons per minute.
Now if we take the case of the second tank, then,
Speed of filling the second tank = (5/8) * 2
= 10/8
So the second tank fills at the rate of 10/8 gallons per minute.
Now we can say that the second tank is getting filled at a faster rate.
