Please answer asap! quadratic equations.

Question

the length of a base of a rectangular prism is 2m more than its width, and the height of the prism is 15m.
a)write an algebraic expression for the volume of the rectangular prism
b) the volume of the prism is 2145m³. Write an equation to model the situation.
c)solve the question in part b) by factoring. What are the dimensions of the base of the prism?

Answer 1

a) A rectangular prism is just calculating a rectangle in three dimensions, so it's length times width times height, or:

V = lwh

We're told the following:

l = 2 + w

h = 15

So the equation looks like:

V = 15w(2 + w)

or

V = 15w² + 30w

 

b) We're given the volume, so our equation now looks like:

2145 = 15w(2 + w)

143 * 15 = 15w(w + 2)

143 = w(w + 2)

143 = w² + 2w

0 = w² + 2w - 143

 

Now we have a quadratic formula. Using what we know, the numbers that multiply together to make 143 have to be only two units apart (so that we can have that 2w in the center), thus very close together. The square root of 143 is almost twelve, so let's try that:

10  * 12 = 120

12* 14 = 168

 

Close, so let's try another pair around that:

11 * 13 = 143

There we go. Now since we have a plus two, that means the larger number gets the addition sign when we factor out:

0 = (w - 11)(w + 13)

So:

0 = w - 11

or:

0 = w + 13

which means w = 11, -13

We can't have a negative length of a side, so the width has to be 11 meters.

 

l = 11 + 2

l = 13

 

Now lets check:

2145 m³ = 11m * 13m * 15m

2145 m³ = 2145m³

 

Now, when the question asks you to find the base, I assume they mean the flat surface on the ground, so:

11m * 13m = 143m²