Evan is making a table that will be created in the shape of the figure below. The table top is a triangle attached to a rectangle. To purchase the right amount of paint, he needs to know the area of the table top. He can only spend $10 on paint, which is enough to cover 150 ft2 of surface area. What is the maximum length of the base of the rectangle he can build?

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Evan is making a table that will be created in the shape of the figure below. The table top is a triangle attached to a rectangle. To purchase the right amount of paint, he needs to know the area of the table top. He can only spend $10 on paint, which is enough to cover 150 ft2 of surface area. What is the maximum length of the base of the rectangle he can build?

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To find the maximum length of the base of the rectangle, we first need to calculate the area of the table top.

The table top is a triangle attached to a rectangle. To find its area, we need to calculate the areas of the triangle and the rectangle separately, and then sum them up.

Let's assume the base of the triangle is the base of the rectangle. Let's call it "x".

The area of the triangle can be calculated using the formula: (base * height) / 2. Since the height is not given, let's assume it is also "x".

Therefore, the area of the triangle is (x * x) / 2, which simplifies to x² / 2.

Now, let's calculate the area of the rectangle. The length of the rectangle is also "x" and let's assume the width is "y". Therefore, the area of the rectangle is x * y.

To find the area of the table top, we need to add the areas of the triangle and the rectangle. So, the total area (A) is given by the equation A = (x² / 2) + (x * y).

Since we know that Evan can only spend $10 on paint, which covers 150 ft² of surface area, we can set up the equation:

A = 150

Substituting the equation for the total area, we get (x² / 2) + (x * y) = 150.

We want to find the maximum length of the base of the rectangle, which is "x". To do this, we need to optimize the equation for A by expressing "y" in terms of "x" and then finding the maximum value of A.

Since we don't have any information about the relationship between "x" and "y", we cannot optimize the equation for A without further information.