Question 927379: Dear Tutor,
Your help in solving this problem is much appreciated. Thank you!
A rectangular field is 300 yards by 500 yards. A roadway of width x yards is to be built inside
the field. This problem concerns the region inside the roadway.
a. Write the length and width of the region as functions of x. What kind of functions are these?
)I found the answer but don't understand why it is so)
L for length(x) = 500-2x
w for width(x) = 300-2x (why 2x instead of x?)
Linear Function
500 (sorry not drawn to scale)
__________________
l l
l _____x_____ l
l l l l
l l l l 300
l l l l
l l l l
l ---------- l
l l
----------------
b. Write the area of the region as a function of x. What kind of function is this?
A (x) = (500-2x)(300-2x)
= 4x^2 -1600x + 150.000
Quadratic
c. Find the value of x which makes the roadway have an area equal to the area of the field.
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! a) the roadway has a width of x, therefore there are 2x to be subtraced from both the length and the width (note that the length and width each have two corners of width x and length x)
b) A (x) = (500-2x)(300-2x)
= 4x^2 -1600x + 150,000
Quadratic, this is a parabola that opens upward
note that the third term is 150,000 not 150.000 ( I think that was just a typo :-) )
c) Area of the rectangular field is 500*300 = 150,000
The quadratic 4x^2 -1600x + 150,000 factors into 4*(x-250)*(x-150), now set this equal to the area of the rectangular field
4*(x-250)*(x-150) = 150,000
(x-250)(x-150) = 37500
x^2 - 400x +37500 = 37500
x^2 -400x = 0
x(x-400) = 0
x is either 0 yards or 400 yards
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