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 6 feet, and the width by 2 feet. If the area of the new rectangular deck is 68 ft2 larger than the area of the original deck, find the dimensions of the original deck.
original deck : x by 4x
original area = 4x^2
new deck:
length = 4x+6
new width = x+2
new area = (x+2)(4x+6)
(x+2)(4x+6) - 4x^2 = 68
expand the left side, simplify and solve for x
comes out very nicely
To solve this problem, let's first assume that the length of the original deck is "L" feet.
According to the problem, the width of the original deck is one-fourth of the length, so the width is (1/4)L feet.
The area of the original deck is given by multiplying the length and width:
Original area = Length * Width = L * (1/4)L = L^2/4 ft^2.
The length of the new deck is expanded by 6 feet, so the new length is (L + 6) feet.
Similarly, the width is expanded by 2 feet, so the new width is (1/4)L + 2 feet.
The area of the new deck is given by multiplying the new length and width:
New area = (L + 6) * ((1/4)L + 2) ft^2.
The problem states that the area of the new deck is 68 ft^2 larger than the area of the original deck. Hence, we can write the following equation:
New area - Original area = 68 ft^2
[(L + 6) * ((1/4)L + 2)] - (L^2/4) = 68
Simplifying this equation will give us the value of L. Let's solve it step by step:
[(L + 6) * ((1/4)L + 2)] - (L^2/4) = 68
[(L + 6) * (1/4)L + 2*(L + 6)] - (L^2/4) = 68
[(L^2 + 6L)/4 + 2L + 12] - (L^2/4) = 68
[(L^2 + 6L + 8L + 48) - (L^2)]/4 = 68
(14L + 48)/4 = 68
14L + 48 = 68*4
14L + 48 = 272
14L = 272 - 48
14L = 224
L = 224/14
L = 16
Therefore, the length of the original deck is 16 feet.
We can also find the width of the original deck using the equation:
Width = (1/4)L
Width = (1/4)*16
Width = 4 feet
So, the dimensions of the original deck are 16 feet by 4 feet.