A carpenter is putting a skylight in a roof. If the roof measures 10𝑥 + 8 by 8𝑥 + 6 and the skylight measures 𝑥 + 5 by 3𝑥 + 4, what is the area of the remaining roof after the skylight is built?
a. 77𝑥^2 + 115𝑥 + 68
b. 77𝑥^2 + 105𝑥 + 28
c. 83𝑥^2 + 115𝑥 + 68
d. 77𝑥^2 − 105𝑥 + 28
To find the area of the remaining roof after the skylight is built, we need to subtract the area of the skylight from the area of the original roof.
The area of the original roof is found by multiplying the length and width:
Original roof area = (10𝑥 + 8) * (8𝑥 + 6)
The area of the skylight is found by multiplying its length and width:
Skylight area = (𝑥 + 5) * (3𝑥 + 4)
To find the area of the remaining roof, we subtract the skylight area from the original roof area:
Remaining roof area = (10𝑥 + 8) * (8𝑥 + 6) - (𝑥 + 5) * (3𝑥 + 4)
Now, let's simplify this expression:
Expanding the first term:
= 80𝑥^2 + 60𝑥 + 64𝑥 + 48
Expanding the second term:
= 3𝑥^2 + 4𝑥 + 15𝑥 + 20
Now, we can combine like terms:
= 80𝑥^2 + 124𝑥 + 48 + 3𝑥^2 + 19𝑥 + 20
Combining like terms again:
= (80𝑥^2 + 3𝑥^2) + (124𝑥 + 19𝑥) + (48 + 20)
= 83𝑥^2 + 143𝑥 + 68
Therefore, the area of the remaining roof after the skylight is built is 83𝑥^2 + 143𝑥 + 68.
So, the correct answer is c. 83𝑥^2 + 115𝑥 + 68.
To find the area of the remaining roof after the skylight is built, we need to subtract the area of the skylight from the area of the original roof.
The area of a rectangle is given by the formula length × width.
The length of the original roof is 10𝑥 + 8 and the width is 8𝑥 + 6, so the area of the original roof is (10𝑥 + 8) × (8𝑥 + 6).
The length of the skylight is 𝑥 + 5 and the width is 3𝑥 + 4, so the area of the skylight is (𝑥 + 5) × (3𝑥 + 4).
To find the area of the remaining roof, we subtract the area of the skylight from the area of the original roof:
(10𝑥 + 8) × (8𝑥 + 6) - (𝑥 + 5) × (3𝑥 + 4)
Next, we can simplify the equation by multiplying out the terms:
(80𝑥^2 + 60𝑥 + 64𝑥 + 48) - (3𝑥^2 + 4𝑥 + 15𝑥 + 20)
Now, let's combine like terms:
80𝑥^2 + 124𝑥 + 48 - 3𝑥^2 - 19𝑥 - 20
Finally, collect like terms and simplify:
77𝑥^2 + 105𝑥 + 28
Therefore, the area of the remaining roof after the skylight is built is 77𝑥^2 + 105𝑥 + 28.
The correct answer is option b. 77𝑥^2 + 105𝑥 + 28.
[(10𝑥 + 8) (8𝑥 + 6)] - [(𝑥 + 5) (3𝑥 + 4)] = ?
(80𝑥^2 + 124𝑥 + 48) - (3𝑥^2 + 19𝑥 + 20)