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Question 157684: Using polynomials, could you write this out for me, with solution?
Linda has a garden. The width of her garden is 8 feet longer than the length of it. Around the garden she has a 4 foot wide sidewalk. The area of the sidewalk is 320 feet squared. What are the dimensions of the garden?
Thank you.
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! draw a rectangle and surround it by a larger rectangle.
let ABCD = smaller rectangle = garden
let PQRS = larger rectangle = outer dimension of sidewalk.
ABCD = inner dimension of sidewalk plus outer dimension of garden.
lettering should be clockwise from top left corner.
for the smaller rectangle, A is top left corner, B is top right corner, C is bottom right corner, D is bottom left corner.
for the larger rectangle, P is top left corner, Q is top right corner, R is bottom right corner, S is bottom left corner.
let AB be the width of the garden.
let BC be the length of the garden.
let PQ be the width of the outer dimension of the sidewalk.
let QR be the length of the outer dimension of the sidewalk.
let area of garden = A(G).
since the area of the garden is enclosed within the outer dimensions of the sidewalk, the area of the sidewalk can be calculated as the area enclosed by the outer dimensions of the sidewalk minus the area enclosed by the inner dimensions of the sidewalk which are also the outer dimensions of the garden. this is useful to simplify the calculations of the area of the sidewalk.
taking this into account, .....
let area of sidewalk = A(S) = A(ODOS) - A(G) where ODOS means "outer dimensions of sidewalk".
You are given that the width of the garden is 8 feet longer than the length of the garden.
that means that AB = BC + 8
you are also given that the sidewalk is 4 feet wide at all points.
that means that the width of the outer dimension of the sidewalk must be equal to the width of the garden + 8 feet (4 on each side).
that means that PQ = (BC + 8) + 8 = BC + 16.
it also means that the length of the outer dimension of the sidewalk must be equal to the length of the garden + 8 feet (4 on each side).
that means that QR = BC + 8.
you should now have all that you need to start calculating.
the area of the garden can be calculated as follows:
area of garden equals length * width, which becomes
A(G) = AB*BC
since AB = BC+8, the formula becomes ....
A(G) = (BC+8)*BC
can't do anything with this now so hold on to it for later when the value of BC becomes known.
area of sidewalk is given as 320 square feet.
this is known as the area enclosed by the outer dimensions of the sidewalk minus the area enclosed by the area of the garden which is the same as the area enclosed by the inner dimensions of the sidewalk.
formula for this is A(S) = A(ODOS) - A(G)
formula for A(ODOS) = length times width of the outer dimensions of the sidewalk.
this becomes ....
A(ODOS) = PQ * QR = (BC+16)*(BC+8)
so formula for A(S) becomes ((BC+16) * (BC+8)) - (BC+8) * BC) = 320
let x = BC.
formula becomes ((x+16)*(x+8)-(x+8)*x)=320
multiplying out, formula becomes...
x^2 + 16*x + 8*x + (16*8) - x^2 - 8*x = 320
doing the addition and subtraction of like terms, the equation becomes...
16*x + 128 = 320
which becomes...
16*x = 192
which becomes...
x=12.
now that we know the value of x, we can get the other dimensions.
if x = 8, then...
BC = 12
AB = BC + 8 = 20
PQ = AB + 8 = 28
QR = BC + 8 = 20
length of the garden = BC = 12.
width of the garden = AB = 20.
these are also the inner length and inner width of the sidewalk.
outer width of the sidewalk = PQ = 28.
outer length of the sidewalk = QR = 20.
outer width of sidewalk is 8 feet more than width of garden = 28 - 20 = 8.
outer length of sidewalk is 8 feet more than length of garden = 20 - 12 = 8.
width of garden is 8 feet more than length of garden = 20 - 12 = 8.
all parameters check out ok.
area of garden is 12 * 20 = 240 square feet
area enclosed by sidewalk is 28 * 20 = 560 square feet.
area of sidewalk is area enclosed by sidewalk minus area of garden = 560 square feet minus 240 square feet = 320 square feet.
everything checks out ok.
dimensions of the garden are 20 feet width and 12 feet length.
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