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Question 489023: Help me please? I don't get this at all....
Essay: Show all work. A school designer wants to create a whiteboard with the optimal dimensions to enhance learning. It is determined that if one side of the whiteboard is x + y inches, the other side should be x^2-3xy+7y^2 inches. Write an algebraic expression for the area of such a whiteboard, simplify it, and include correct units with your solution.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! one side of the white board is x + y inches.
the other side of the white board is x^2 - 3xy + 7y^2 inches.
the area of the whiteboard is equal to length * width.
let W equal the width and set W equal to x + y inches.
let L equal the length and set L equal to x^2 - 3xy + 7y^2 inches.
let A equal the area which is equal to L * W.
you get:
A = L * W = (x + y) * (x^2 - 3xy + 7y^2) square inches.
you need to multiply these factors together.
(x + y) * (x^2 - 3xy + 7y^2) will be equal to:
(x * (x^2 - 3xy + 7y^2)) + (y * (x^2 - 3xy + 7y^2)) which becomes equal to:
(x^3 - 3x^2y + 7xy^2) + (yx^2 - 3xy^2 + 7y^3)
remove the parentheses and reorder the variables in each term so that the x variable is first and the y variable is second so you can see the like terms easier.
you get:
x^3 - 3x^2y + 7xy^2 + x^2y - 3xy^2 + 7y^3
now reorder the terms so that the like terms are next to each other.
you get:
x^3 [- 3x^2y + x^2y] [+ 7xy^2 - 3xy^2] + 7y^3
i enclosed the like terms in brackets so you can see them better.
you would not ordinarily do that.
i was careful to include the operation signs so you can see whether they needed to be added together or subtracted from each other.
now combine like terms to get:
x^3 -2x^2y + 4xy^2 + 7y^3
that's your area.
it is expressed in square inches.
you would say:
A = x^3 - 2x^2y + 4xy^2 + 7y^3 square inches.
if you want to confirm that you did the calculations correctly, then the easiest way is to provide some random values for x and y and use them to solve the original equation and the final equation after simplification.
the original equation is:
A = (x + y) * (x^2 - 3xy + 7y^2) square inches.
the final equation is:
A = x^3 - 2x^2y + 4xy^2 + 7y^3 square inches.
i did the check using x = 2 and y = 3.
i got an answer of 245 both times so i'm reasonably confident in did the calculations and the simplifications correctly.
the multiplication uses the distributive property of mathematics.
(a + b) * (c + d) would be equal to:
a * (c + d) + b * (c + d)
a + b) * (c + d + e) would be equal to:
a * (c + d + e) + b * (c + d + e)
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