The volume of a rectangular prism is 300 cubic inches. The length of the rectangular prism is 4 inches,
the width is x + 4, and the height is x – 6.
Find the value of x, and the dimensions of the rectangular prism.
V = lwh

x=?
Width =?
Height=?

well I dont really know but I got this 8x-6

Step-by-step explanation:

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.