A carpenter is assigned the job of expanding a rectangular deck where the width is one-fourth the length. The length of the deck is to be expanded by 10 feet, and the width by 6 feet. If the area of the new rectangular deck is 128 ft^2 larger than the area of the original deck, find the dimensions of the original deck.
Length = L1.
Width = L1/4.
L2 = L1+10.
W2 = L1/4 + 6.
A2 = A1 + 128 Ft^2.
A2 = (L1*L1/4) + 128.
L2*W2 = (L1*L1/4) + 128.
(L1+10)*(L1/4) + 6) = (L1*L1/4)+128.
Multiply both sides by 4:
(L1+10)*(L1)+24 = (L1*L1) + 512.
L1^2+10L1+24 = L1^2 + 512.
L1^2 - L1^2 + 10L1 + 24 = 512.
10L1 = 512-24 = 488.
L1 = 48.8 Ft.
W1 = L1/4 = 48.8/4 = 12.2 Ft.
To solve this problem, we need to use the information given and set up an equation based on the given conditions.
Let's assume the original length of the deck is "L" feet. According to the problem, the width of the deck is one-fourth of the length, which means the original width is (1/4)L feet.
The area of a rectangle can be calculated by multiplying its length by its width. So, the area of the original deck is:
Area of original deck = Length * Width = L * (1/4)L = (1/4)L^2 square feet
Now, it's given that the length of the deck is expanded by 10 feet, so the new length is (L + 10) feet. Similarly, the width is expanded by 6 feet, so the new width is ((1/4)L + 6) feet.
The area of the new rectangular deck is given to be 128 square feet larger than the area of the original deck. So, we can set up the following equation:
Area of new deck - Area of original deck = 128
Replace the area values with their respective formulas:
[(L + 10) * ((1/4)L + 6)] - [(1/4)L^2] = 128
Simplify the equation:
[(L^2 + 10L) * (L/4 + 6)] - [(L^2)/4] = 128
Now, we need to solve this equation to find the value of L, which is the original length of the deck.
To do that, we can multiply both sides of the equation by 4 to eliminate the fraction:
4[(L^2 + 10L) * (L/4 + 6)] - [(L^2)/4] * 4 = 128 * 4
(L^2 + 10L) * (L + 24) - L^2 = 512
Expand the product:
(L^3 + 24L^2 + 10L^2 + 240L) - L^2 = 512
Combine like terms:
L^3 + (24L^2 + 10L^2 - L^2) + 240L = 512
Simplify further:
L^3 + 33L^2 + 240L = 512
Rearrange the equation to bring all terms to one side:
L^3 + 33L^2 + 240L - 512 = 0
Now, we have a cubic equation in terms of L. We can solve this equation using numerical methods such as graphing, factoring, or using a calculator or computer program to find the value(s) of L.
Once we find the values of L, we can substitute them back into the original equations to find the corresponding widths and verify the solution.
Note: The exact values of L and the corresponding dimensions may involve decimals or fractions, but the above steps should help you in finding the dimensions of the original deck.