Answer:
x = 11
width = 15
height = 5
Step-by-step explanation:
Volume = length * width * height
Given:
Volume = 300
Length = 4
Width = x + 4
Height = x - 6
If volume = length * width * height
Then 300 = 4(x+4)(x-6)
This is the equation we will use to solve for x
300 = 4(x+4)(x-6)
Divide both sides by 4
75 = (x+4)(x-6)
Next use FOIL to expand the factor ( see down below in the notes section to see how to do it if you don't know how )
After FOILing we get x² - 2x - 24 = 75
There are many methods we could use to solve for x from here but I'm going to solve it by completing the square.
x² - 2x - 24 = 75
The first step is to get the x terms by themselves.
To do this we want to get rid of the -24.
We can do this by using inverse operations and adding 24 to both sides.
x² - 2x - 24 = 75
add 24 to both sides
x² - 2x = 99
Next we want to "complete the square".
We do this by squaring half of the x term (-2)
(-2/2)² = 1
We then add the result (1) to both sides
x² - 2x + 1 = 100
Next we factor the left side rewrite the perfect square as (x-1)²
(x-1)² = 100
We then take the square root of both sides
√(x-1)² = √100
We acquire x-1 = +-10
We the solve for x two times, once when the 10 is positive and once when the 10 is negative
First solution x - 1 = 10
add 1 to both sides
x = 11
Second solution x - 1 = -10
add 1 to both sides
x = -9
If x = the value of a side length then we know that it must be positive because we cannot have a negative side length therefore the solution is x = 11
We now find the length of the width and height
Width = x + 4
x = 11
width = 11 + 4 = 15
Height = x - 6
x = 11
height = 11 - 6 = 5
We can check our work to make sure
Volume = length * width * height
300 = 5 * 15 * 4
5 * 15 * 4 = 300
300 = 300 ✓
And we are done!
Notes:
FOIL stands for front, outer, inner and last.
The FOIL method is used to make sure that the factor is expanded correctly and used to avoid missing out on multiplying 2 terms.
We used FOIL in step 2 to expand the factor and these are the steps
(x + 4)(x-6)
First multiply the front terms x and x to get x²
Then multiply the outer terms -6 and 4 to get -24
Then multiply the inner terms 4 and x to get 4x
Finally multiply the "last" terms -6 and x to get -6x
We then add all of the products up to get x² - 6x + 4x - 24
Lastly, we combine like terms to get x² - 2x - 24
We could have also solved using the quadratic formula (-b + or - √b² - 4(a)(c))/2(a)
The quadratic formula always works for a quadratic equation.
