Question

Optimization: Find the base and height of the largest rectangle which can be drawn

within the region bounded above by y-4-x², and below by the x-axis, if the bottom of
edge of the rectangle should be on the x-axis.

Answer 1

To find the base and height of the largest rectangle within the given region, set y = 0 and solve for x to find the x-values where the top of the rectangle intersects the curve y = 4 - x². Calculate the corresponding y-values at these x-values to determine the maximum height of the rectangle. The base length is 2 times the absolute value of the x-value where the top of the rectangle intersects the curve.

Step-by-step explanation:

To find the base and height of the largest rectangle within the given region, we need to first determine the x-values at which the top of the rectangle intersects the curve y = 4 - x². Since the bottom edge of the rectangle lies on the x-axis, the y-coordinate of its top must be 0. Setting y = 0, we get 4 - x² = 0. Solving this equation, we find x = ±2.

Next, we calculate the corresponding y-values of the curve at x = 2 and x = -2. These values give us the maximum height of the rectangle. Evaluating the curve at x = 2, we get y = -4, and at x = -2, we get y = 0. Therefore, the height of the rectangle is 4 units.

Finally, we can calculate the base of the rectangle. Since the bottom edge of the rectangle lies on the x-axis, the base length is simply 2 times the absolute value of the x-value where the top of the rectangle intersects the curve. Hence, the base length is 2 * |2| = 4 units.

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