Question 1114462: Two students disagree about how to solve a problem so they ask you for help. They have 220 ft of ribbon to section off a rectangular space in
the gymnasium for a dance floor, but cannot decide on the dimensions. The
dance floor has to be at least 70 ft long to accommodate the anticipated
number of dancers.
Write a system of inequalities. Then draw a graph that shows all of the
possible dimensions of the dance floor.
Write two possible solutions to the problem that would be appropriate
dimensions for a dance floor. Compare your solutions and determine
which would offer more dancing space. Explain your reasoning.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the minimum length is 70 feet.
the maximum width is therefore 40 feet.
this is because the maximum perimeter is 220 feet.
perimeter = 2 * length + 2 * width.
this becomes 220 = 2 * length + 2 * width.
divide both sides of this equation by 2 to get:
110 = length + width.
length has to be greater than 70, therefore width has to be less than or equal to 40.
under the assumption that you want to use the maximum amount of ribbon available, you get:
220 = 2 * (L + W)
divide both sides of this equation by 2 to get:
110 = L + W, where L is the length and W is the width.
solve for W to get W = 110 - L.
the area of the dance floor is L * W.
therefore A = L * W, where A is the area, L is the length and W is the width.
since W = 110 - L, this equation becomes A = L * (110 - L).
simplify to get A = 100L - L^2.
arrange the equation in order of degree to get:
A = -L^2 + 100L
L will be >= 70
replace A with y and L with x and you have the following inequalities.
y <= -x^2 + 100x
x >= 70
y >= 0
using the desmos.com calculator, you would graph the opposite of these inequalities.
you would graph.
y >= -x^2 + 100x
x <= 70
y <= 0
the area of the graph that is not shaded is your region of feasibility.
the graph looks like this:
the graph shows that the maximum area is 2800 square feet when x = 70
when x = 70, y must be equal to 40, because x represents the length and length * width = area, and 70 * 40 = 2800.
you have a maximum area of the dance floor when the length is 70 feet and the width is 40 feet.
you can also see fromt he graph that the maximum area of the dance floor could have been 3025 square feet if the length could have been less than 70 feet, but that is not possible because of the requirement for the length to be greater than or equal to 70 feet.
a length of 55 feet puts x into the shaded area of the graph which is not in the feasible region.
your maximum area is when the length is equal to 70 and the width is equal to 40.
any other length or width will be less than that.
for example, when the length is 90, the width has to be 110 - 90 = 20 and length * width becomes 90 * 20 = 1800, which is less than the maximum.
|
|
|
