To find the lengths of the edges of a right square-based prism where the height is twice the length of the base edge, and given the surface area is 49, you can follow these steps:
1. Start by understanding the formula for the surface area of a right square-based prism. The surface area (SA) is calculated by adding the areas of all six faces, where the base is a square and there are four rectangular side faces. The formula is as follows:
\[
SA = 2 \cdot \text{area of base} + \text{perimeter of base} \cdot \text{height}
\]
2. Since the base is a square, let's denote the length of the base edge as \(x\). The area of the base is then \(x^2\), and the perimeter of the base is \(4x\).
3. Given that the height (\(h\)) of the prism is twice the length of the base edge, we can write \(h = 2x\).
4. You can now express the formula for the surface area in terms of \(x\):
\[
SA = 2 \cdot x^2 + 4x \cdot 2x
\]
Simplify the formula:
\[
SA = 2x^2 + 8x^2 = 10x^2
\]
5. Since the total surface area is given to be 49, we equate \(10x^2\) to 49:
\[
10x^2 = 49
\]
6. Divide both sides of the equation by 10 to solve for \(x^2\):
\[
x^2 = \frac{49}{10}
\]
\[
x^2 = 4.9
\]
7. Find the length of the base edge \(x\) by taking the square root of 4.9:
\[
x = \sqrt{4.9}
\]
Note: The length of an edge must be a positive number because you cannot have a negative physical length.
8. Use a calculator to find the square root of 4.9:
\[
x \approx 2.2136
\]
9. Now we can find the height of the prism, which is twice the length of the base edge. Let's call it \(h\):
\[
h = 2x \approx 2 \cdot 2.2136 \approx 4.4272
\]
So, the length of the base edge \(x\) of the prism is approximately 2.2136 units, and the height \(h\) of the prism is approximately 4.4272 units.
