step by step
A. Express BC in terms of b:
BC is the length between the points B and C on the curve y = 4 - x^2. Since A and B lie on the axis, B has coordinates (b, 0). To find the y-coordinate of point C, we substitute the x-coordinate b into the equation of the curve:
y = 4 - x^2
Plugging in x = b, we get:
y = 4 - b^2
Therefore, point C has coordinates (b, 4 - b^2). To find the length BC, we subtract the y-coordinate of B (0) from the y-coordinate of C (4 - b^2):
BC = (4 - b^2) - 0 = 4 - b^2 units
B. Show that A = 2p(4 - p^2):
The area of the rectangle ABCD is given by the formula: A = length × width.
The length of the rectangle, AB, is equal to the base length of the rectangle, which is given as 2p units.
The width of the rectangle, BC, can be found using the result from part A, which is BC = 4 - b^2 units.
Multiplying the length AB by the width BC, we get:
A = (2p) × (4 - b^2)
A = 8p - 2pb^2
A = 2p(4 - b^2)
Therefore, A = 2p(4 - b^2) units^2.
C. Find the value of p for which A has a stationary value:
To find the value of p for which A has a stationary value, we need to find the critical points where the derivative of A with respect to p equals zero.
We differentiate A = 2p(4 - p^2) with respect to p:
dA/dp = 2(4 - p^2) - 2p(2p)
dA/dp = 8 - 2p^2 - 4p^2
dA/dp = 8 - 6p^2
Now, we set dA/dp equal to zero and solve for p:
8 - 6p^2 = 0
6p^2 = 8
p^2 = 8/6
p^2 = 4/3
p = ± √(4/