Question

Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola.

y=5-x²

Answer 1

To find the dimensions of the rectangle with the largest area, we need to find the coordinates of the two vertices above the x-axis on the parabola y = 5 - x².

One vertex is (√5, 0) on the x-axis, and the other is (-√5, 0).

Therefore, the dimensions of the rectangle are (2√5, 0).

Step-by-step explanation:

The given equation of the parabola is y = 5 - x².

Let's substitute y = 0 to find the x-coordinate of the vertex on the x-axis.

0 = 5 - x² ⟹ x² = 5

⟹ x = ±√5.

Since we want the vertex above the x-axis, we take x = √5.

Substituting this value back into the equation, we find that the y-coordinate is 5 - (√5)²

= 5 - 5

= 0.

So, one vertex of the rectangle is (√5, 0) on the x-axis. The other vertex must have the same y-coordinate and a positive x-coordinate.

Since rectangles have opposite sides equal, the other vertex will be (-√5, 0).

The dimensions of the rectangle are the difference between the x-coordinates and the y-coordinate.

Therefore, the dimensions are (√5 - (-√5), 0 - 0) which simplifies to (2√5, 0).