Part 1: You finally made it…graduation day has arrived! Your parents are so proud that they are throwing you a HUGE party. They have rented a big tent for your graduation party just in case it rains. The inside of the tent (represented by the blue area) is set up for a dance floor for you and friends to dance the night away.
The rectangular dance floor is twice as long as it is wide. The tent surrounds the dance floor, leaving some space (7 feet in each direction) for guests to mingle and cool off after dancing. This extra space (represented by the grey area) has an area of 952 square feet. Your task is to find the dimensions of the dance floor. Be sure to show all work.
952 = (2W + 14) * (W + 14)
2W^2 + 28W - 756 = 0
2(W^2 + 14W - 378) = 0
W=sqrt(427)-7
To find the dimensions of the dance floor, we can set up an equation using the given information.
Let's assume the width of the dance floor is x feet. Since the length is twice the width, the length would be 2x feet.
The area of the dance floor can be found by multiplying the length and the width:
Area of the dance floor = length × width = (2x) × x = 2x^2 square feet.
Now, we know that the extra space surrounding the dance floor has an area of 952 square feet. The extra space is represented by the grey area, which is a rectangle. We can find the area of this rectangle by subtracting the area of the dance floor from the total area of the tent.
Total area of the tent = Area of the dance floor + Area of the extra space
Total area of the tent = 2x^2 + 952 square feet.
It is given that the total area of the tent is represented by the blue area, which is a rectangle. The length of this rectangle would be 2x + 14 feet (adding 7 feet on each side of the dance floor) while the width would be x + 14 feet (adding 7 feet on each side of the dance floor).
We can now set up an equation using the area of the total tent area:
Total area of the tent = length × width = (2x + 14) × (x + 14) square feet.
Now we can solve for x by setting the two equations for the total area of the tent equal to each other:
2x^2 + 952 = (2x + 14) × (x + 14).
By multiplying out the right side and simplifying, we get:
2x^2 + 952 = 2x^2 + 28x + 196.
Now, we subtract 2x^2 from both sides to get rid of the quadratic term:
952 = 28x + 196.
Subtracting 196 from both sides gives us:
756 = 28x.
Finally, we divide both sides by 28:
x = 756/28 = 27.
Therefore, the width of the dance floor is 27 feet, and the length is 2x = 2(27) = 54 feet.