Rectangles beneath a line

a. A rectangle is c...

Question

Rectangles beneath a line

a. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area?

Accepted Answer

A rectangle is positioned in the first quadrant of a coordinate plane with one vertex at the origin, 00, 1 side along the positive X axis, another along the positive Y axis, and its opposite vertex on the line Y equals 8 minus X. Determine the dimensions of the rectangle that maximize the area and find its maximum area. Now, to solve this, let's first draw a diagram. We have a point at 00. You have a point on the X axis, we'll call this X0. One point on the Y axis, 0 Y. At another point along the line. Y equals 8 minus X. Now, to solve this, let's first look at the area of a rectangle. If a rectangle is given by length multiplied by width. Or in our case, X multiplied by Y. Now, note that Y equals 8 minus X. We can make a substitution by saying X is multiplied by 8 minus X. We get A equals 8 X minus X2. Now, to maximize this, we will take the first derivative and set this equal to 0. We'll say A equals 0. By saying A equals 8 minus 2 X. If we set a equal to 0, We can solve for X. Subtract 8. To get 8 equals -2 X. And we divide We get X equals 4. Now, we can check if this is the maximum by taking the 2nd derivative test. So we will then say a double prime. This equals to the next derivative. -2. Because this value is less than 0. This is a maximum. Now let's find the way. Y is equals to 8 minus X, which is 8 minus 4. That would just be 4. Finally, we can find it in our area. Area is X multiplied by Y. Which is 4, multiplied by 4. Which is 16. We now have our answers. X equals 4. Units Y equals 4 units. An area equals 16. Units squared. OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

Rectangles beneath a linea. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area?

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Rectangles beneath a line

a. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area?