Joe Goodnuff had been eating on a square card table until he bought a new rectangular dining table. The new table is 2 ft longer and 1 ft wider than the card table. It has an area of 24.75 ft². How long was a side of his old square card table?
Please explain this to me; don't just give me an answer!
Answer (3)
The side length of Joe's old square card table was 3.5 feet.
Let's assume the side length of the square card table is x feet. Then, the area of the square table is x^2 square feet.
After Joe bought the new rectangular dining table, the area is given as 24.75 square feet. We can set up the following equation to represent this:
(x + 2) * (x + 1) = 24.75
Expanding and rearranging, we get:
x^2 + 3x + 2 = 24.75
x^2 + 3x - 22.75 = 0
Now, we can solve for x by factoring or using the quadratic formula. After solving, we find that x is approximately 3.5 feet.
So, the side length of Joe's old square card table was 3.5 feet.
The length of a side of Joe Goodnuff's old square card table is 3.5 feet, calculated from the area of the new rectangular table.
Let's denote the length of the side of the old square card table as x feet.
Given that the new rectangular dining table is 2 feet longer and 1 foot wider than the old square card table, its dimensions are (x + 2) feet by (x + 1) feet.
The area A of a rectangle is given by the formula:
A = length × width
Given that the area of the new rectangular table is 24.75 sq. ft, we can set up the equation:
( x + 2 ) ( x + 1 ) = 24.75
Expanding and simplifying this equation:
x 2 + 3 x + 2 = 24.75 x 2 + 3 x + 2 − 24.75 = 0 x 2 + 3 x − 22.75 = 0
Now, we can solve this quadratic equation for x using the quadratic formula:
x = 2 a − b ± b 2 − 4 a c
Where a = 1, b = 3, and c = -22.75.
x = 2 ( 1 ) − 3 ± ( 3 ) 2 − 4 ( 1 ) ( − 22.75 ) x = 2 − 3 ± 9 + 91 x = 2 − 3 ± 100 x = 2 − 3 ± 10
The solutions are:
x 1 = 2 − 3 + 10 = 2 7 = 3.5 ft
x 2 = 2 − 3 − 10 = 2 − 13 = − 6.5 ft
Since the length of a side cannot be negative, we discard the negative solution.
Therefore, the length of a side of Joe Goodnuff's old square card table is \boxed{3.5 \text{ ft}} \
The side length of Joe's old square card table is 3.5 feet. We arrived at this by setting up an equation based on the dimensions of his new rectangular table and solving a quadratic equation. The calculations show that only the positive solution is valid for a physical length.
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