SOLUTION: a park is installing a rectangular reflecting pool surrounded by a concrete walkway of uniform width. the reflecting pool will measure 42 ft by 26 ft. there is enough concrete to c
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Question 854488: a park is installing a rectangular reflecting pool surrounded by a concrete walkway of uniform width. the reflecting pool will measure 42 ft by 26 ft. there is enough concrete to cover 460 ft^3 for the walkway. what is the maximum width of the walkway?
how can drawing a diagram help you solve this problem
how can you write an expression in terms of x for the area of the walkway
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
NOTE:
The problem should state that " there is enough concrete to cover 460 ft^2 for the walkway," because measure of the surface to be covered has to be in square feet. Cubic feet is a measure of volume, and if you have you should ask how deep they want the walkway. Do they want the concrete slabs to be 2 inches thick? Thicker than that?
Drawing not to scale:
How can drawing a diagram help you solve this problem?
A drawing will help you realize that
the reflecting pool is a rectangle (blue in my drawing);
there is a bigger rectangle formed by the walkway plus the reflecting pool (red in my drawing), and
the surface area of the walkway is the difference of the areas of the two rectangles.
You would also notice that if
= uniform width of the walkway (in ft),
the length and width (in ft) of the larger rectangle will be
and .
How can you write an expression in terms of x for the area of the walkway?
You can express the area (in square feet) of the smaller rectangle as .
You can express the area (in square feet) of the bigger rectangle as
=4x^2+84x+52x+42*26=4x^2+136x+42*26}}} .
You can express the area (in square feet) of the walkway as
What is the maximum width of the walkway?
You need to make that area equal or smaller than (460 square feet).
So the maximum will happen when
<-->
Dividing both sides of the equal sign (everything) by 4, we get
Applying the quadratic formula we find
= about
We discard the negative solution, , and are left with
-->
So the maximum width of the walkway is 3.1 feet.
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