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Problem 439

# A rectangle, the lengths of whose base and altitude are in the ratio $$4: 1$$, is equivalent to a square, the length of whose side is 6 inches. Find the dimensions of the rectangle.

Expert verified
The dimensions of the rectangle are 12 inches by 3 inches.
See the step by step solution

## Step 1: Assign variables to the dimensions of the rectangle

Let the base of the rectangle be 4x, and the altitude be x. We are given that the ratio of the dimensions is 4:1, so this ensures that the ratio is maintained.

## Step 2: Write the equation for the area of the rectangle

The area of the given rectangle is the product of its base and altitude: Area of the rectangle = base * altitude = (4x) * (x) = 4x^2

## Step 3: Write the equation for the area of the square

The given square has side length of 6 inches. The area of the square is the side length squared: Area of the square = side^2 = 6^2 = 36 square inches.

## Step 4: Set the area of the rectangle equal to the area of the square and solve for x

Since the area of the rectangle is equivalent to the area of the square, we can set their areas equal and solve for x: 4x^2 = 36 Now, divide both sides by 4 to solve for x^2: x^2 = 9 Take the square root of both sides to find x: x = 3

## Step 5: Find the dimensions of the rectangle

Now that we know the value of x, we can find the dimensions of the rectangle: Base = 4x = 4 * 3 = 12 inches Altitude = x = 3 inches The dimensions of the rectangle are 12 inches by 3 inches.

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