Word problem

Question

Mrs. Smith has asked Mr. Smith to build a sandbox for their two children Emily and John with the following proportions with the following budget constraints. The base of the sandbox is to be square with a side of length x+5 inches and a height of 2x-6 inches. How many cubic inches of sand would be needed to fill the box? Mrs. Smith insists that the sandbox be covered with chicken wire so that no animals can get in the sandbox. What is the minimum amount of chicken wire that must be purchased to cover the sandbox?

Answer 1

Hi Victoria,

 

The volume of wire required will be equal to the volume of the sandbox.

 

Therefore, our answer will be the product of the dimensions of the sandbox:

    (x + 5)²(2x-6)

 

Let's first find the product of (x + 5)² by using FOIL to multiply these binomials:

    (x + 5)² = (x + 5)(x + 5)

 

    (x + 5)(x + 5) = (x)(x) <F>irst terms +

                           (x)(5) <O>utside terms +

                           (5)(x) <I>nside terrms +

                           (5)(5) <L>ast terms

    (x + 5)² = x² + 10x + 25

 

Finally, we'll multiply the height of the sandbox by the result of squaring its base (x + 5)^2::

    (2x - 6)(x² + 10x + 25)

 

Multiplying the 1st term of the binomial by each term of the trinomial, we get:

    (2x)(x²) + (2x)(10x) + (2x)(25) 

    2x³ + 20x² + 50x (1st partial answer)

 

Multiplying the 2nd term of the binomial by each term of the trinomial, we get:
    (-6)(x²) + (-6)(10x) + (-6)(25)
    -6x² - 60x - 150 (2nd partial answer)

 

Finally, combining both partial answers, we get

       2x³ + 20x² + 50x

  + (       -   6x² -  60x - 150 )

  -----------------------------------

       2x³ + 14x² - 10x - 150

      (in³ of chicken wire needed to cover sandbox)

 

Carefully taking these operations step-by-step eventually gives our answer.

 

Thanks for submitting this question and glad to help.

 

God bless, Jordan.