Find k so that the distance from (–1, 1) to (2, k) is 5 units.

k = -3 or 5

Step-by-step explanation:

The given parameters are;

The line extends from points (-1, 1) to point (2, k)

The length of the line = 5 units

The formula for the length, l, of a line given its coordinates can be  found by the following formula;

Therefore, we have;

Which, by squaring both sides, gives;

25 = (k - 1)² + (2 - (-1))² = (k - 1)² + (2 + 1)² = (k - 1)² + 3²

25 = (k - 1)² + 3² = k² - 2·k + 1 + 9

25 - 25 = 0 = k² - 2·k + 1 + 9 - 25 = k² - 2·k - 15

0 = k² - 2·k - 15

0 = (k +3) × (k - 5)

Therefore, k = -3 or 5

When k - -3, we have;

When k = 5, we have;

The dimensions of the rectangle that maximizes the area is 24 inches by 24 inches.

Step-by-step explanation:

We are given that a metallic bar with a length of 96 inches was used to make a rectangular frame.

And we want to find the dimensions of the frame that yields the maximum area.

Since the bar has a length of 96 inches, the perimeter of the rectangle must total 96. Let l represent the length of the rectangle and w its width. Hence:

Divide:

The area of the rectangle will be:

From the first equation, solve for either variable:

Substitute:

Distribute:

Since the area of the rectangle is represented with a quadratic equation with a negative leading coefficient, the maximum area will occur at its vertex point.

The vertex of a quadratic is given by:

In this case, a = -1, b = 48, and c = 0.

Then the length for which the maximum area occurs is:

And the width will be:

In conclusion, the dimensions of the rectangle that maximizes the area is 24 inches by 24 inches.