If the width and height of a rectangular prism are each shrunk to one seventh of the original size but the length remains the same, what is the formula to find the modified surface area?
original length -- x
original width --- y
original height -- z
new length ---- x
new width ----- y/7
new height --- z/7
original SA = 2xy + 2xz + 2yz
new SA = 2x(y/7) + 2x(z/7) + 2(y/7)(z/7)
simplify to whatever version you need
To find the modified surface area of the rectangular prism after shrinking the width and height to one-seventh of the original size while keeping the length the same, you can use the following formula:
Modified Surface Area = 2(lw/49 + lh/49 + wh)
Here's how to derive this formula:
1. Start with the original formula for the surface area of a rectangular prism: Surface Area = 2lw + 2lh + 2wh.
2. Since both the width and height are shrunk to one-seventh of the original size, their new values can be found by dividing the original width (w) and height (h) by 7. Therefore, the new width (w') would be w/7 and the new height (h') would be h/7.
3. The length (l) remains unchanged, so it remains as l.
4. Replace the original width (w) and height (h) in the surface area formula with their modified values: Surface Area = 2(l(w/7) + l(h/7) + wh).
5. Simplify the equation: Surface Area = 2lw/7 + 2lh/7 + 2wh.
6. Factor out 2/7 from each term: Surface Area = (2/7)(lw + lh + 3wh).
7. Multiply both sides by 7/2 to remove the fraction: Modified Surface Area = 2lw + 2lh + 3wh.
So, the modified formula for the surface area of the rectangular prism is:
Modified Surface Area = 2lw + 2lh + 3wh.
Note: In the above explanation, we assume that the length remains the same and only the width and height are modified.