graph y=9-x^2...? A door is to be constructed inside an arch in the shape of the parabola with graph y=9-x^2, where x and y are in meters. If the area of the door is 20 square meters, find the demsion of the door.
graph y=9-x^2...? A door is to be constructed inside an arch in the shape of the parabola with graph y=9-x^2, where x and y are in meters. If the area of the door is 20 square meters, find the demsion of the door.
Every point on the parabolic arch obeys the rule y = 9-x^2. The door will be rectangular and be symmetrical about the y-axis.
If we take any point call it (x1,y1) in the first quadrant as the top right corner of our door it must be on the parabola and thus obeys the same rule.
The y1-value will equal the doors height (L). The x1-value will equal half the doors Breadth (B).
The area of the door will then be Area = L x 2B = (y)(2x). We will leave out the sub-script just for ease of reading!
But y = 9 - x^2 so by substituting we get Area of door = (9-x^2)(2x) =
We are told area = 20 so we get equation : 18x - 2x^3 = 20
Rearranging: 2x^3 - 18x +20 = 0
Solving this we get x=2, x= -0.377 and x = -6.6. The only valid solution is x= 2. subbing back in we get y = 9-2^2 = 5 Door dimensions = 5m high by 4m wide. Big Door!!