word problem

Question

i need step by step process on how to do this problem 

 

the amount of water used to take a shower is directly proportional to the square of time that the shower is used. A shower lasting 6 minutes requires 9 gallons of water. Find the amount of water used in a shower lasting 4 minutes.

Answer 1

First, recognize which type of word (story) problem this is.  It has to do with proportions.  So, look for proportions that are the same, or double, or add to a certain amount, etc.  This problem has a constant proportion:  

   (the amount of water)per(the square of the time) = constant.

We want to find the amount of water, x, used in 4 minutes.

The two observations are that (be sure to square the amount of time):
      (9 gallons / 36 minutes-squared) = (x gallons / 16 minutes-squared) = Constant

We don’t need to know the constant since the first equation gives x.  Let’s solve it:
          9 / 36 = x / 16
          x = (9/36)*16
          x = 4 gallons       (note: minutes-squared cancelled out)

Answer 2

Let x = time the shower is used

Let y = amount of water used

 

 

x and y have a relationship because the amount of water used depends of the time.  We can write a proportion to describe this relationship.

 

y ∝ x²    --->  y increases as x increases

 

 

If we add some constant K, we can establish an equation.

 

y = Kx²

 

Plug in the initial values of x and y into the equation to find K.

 

x = 6

y = 9

 

K = y / x²

K = 9 / (6)²

K = 0.25

 

 

Now, to find how much water is used in 4 minutes, we plug in these values into the equation.

 

K = 0.25

x = 4

 

y = Kx²

y = 0.25(4)²

y = 4

 

 

4 gallons of water is used in a shower lasting 4 minutes.

 

Answer 3

Since it's a direct proportion (variation) , then gal = k(time)^2

 

9 = k(6)^2 = 36 k

k = 9/36 = 1/4

 

4 = 1/4 t^2,

so t^2 = 16,

 

t= 4 MIN ANSWER