can anyone be able to me with this question rq
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
Area of the roof:
Ar = ( 10 x + 8 ) ∙ ( 8 x + 6 ) = 8 x ∙ 10 x + 8 x ∙ 8 + 6 ∙ 10 x + 6 ∙ 8 =
80 x² + 64 x + 60 x + 48 = 80 x² + 124 x + 48
Area of the skylight:
As = ( x + 5 ) ∙ ( 3 x + 4 ) = 3 x ∙ x + 3 x ∙ 5 + 4 ∙ x + 4 ∙ 5 =
3 x² + 15 x + 4 ∙ x + 20 = 3 x² + 19 x + 20
Area of the remaining roof after the skylight is built:
A = Ar - As = 80 x² + 124 x + 48 - ( 3 x² + 19 x + 20 ) =
80 x² + 124 x + 48 - 3 x² - 19 x - 20 =
80 x² - 3 x² + 124 x - 19 x + 48 - 20 =
77 x² + 105 x + 28
Answer b
Well, the area of the remaining roof is essentially the area of the original roof minus the area of the skylight. To find the area of a rectangle, you simply multiply the length by the width. So, let's calculate!
The area of the original roof is (10𝑥 + 8)(8𝑥 + 6), and the area of the skylight is (𝑥 + 5)(3𝑥 + 4).
Now, let's put on our math hats and do some multiplication magic:
(10𝑥 + 8)(8𝑥 + 6) = 80𝑥^2 + 68𝑥 + 48𝑥 + 48 = 80𝑥^2 + 116𝑥 + 48
(𝑥 + 5)(3𝑥 + 4) = 3𝑥^2 + 4𝑥 + 15𝑥 + 20 = 3𝑥^2 + 19𝑥 + 20
Now, subtract the area of the skylight from the area of the original roof:
(80𝑥^2 + 116𝑥 + 48) - (3𝑥^2 + 19𝑥 + 20) = 80𝑥^2 + 116𝑥 + 48 - 3𝑥^2 - 19𝑥 - 20
Combining like terms, we get:
(80𝑥^2 - 3𝑥^2) + (116𝑥 - 19𝑥) + (48 - 20) = 77𝑥^2 + 97𝑥 + 28
Nope, none of the options given match this result.
I guess the remaining area of the roof is mysterious, like a magician's disappearing act. Keep searching, my friend!
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.
1. Start by finding the area of the original roof:
Area of the original roof = length * width
= (10𝑥 + 8) * (8𝑥 + 6)
= 80𝑥^2 + 68𝑥 + 48𝑥 + 48
= 80𝑥^2 + 116𝑥 + 48
2. Next, find the area of the skylight:
Area of the skylight = length * width
= (𝑥 + 5) * (3𝑥 + 4)
= 3𝑥^2 + 4𝑥 + 15𝑥 + 20
= 3𝑥^2 + 19𝑥 + 20
3. Subtract the area of the skylight from the area of the original roof to find the area of the remaining roof:
Area of the remaining roof = Area of the original roof - Area of the skylight
= (80𝑥^2 + 116𝑥 + 48) - (3𝑥^2 + 19𝑥 + 20)
= 80𝑥^2 + 116𝑥 + 48 - 3𝑥^2 - 19𝑥 - 20
= 77𝑥^2 + 97𝑥 + 28
Therefore, the area of the remaining roof after the skylight is built is 77𝑥^2 + 97𝑥 + 28.
So the correct answer is 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 a rectangle is found by multiplying the length by the width. In this case, the length of the roof is given as 10𝑥 + 8 and the width is given as 8𝑥 + 6. So, the area of the original roof would be (10𝑥 + 8) * (8𝑥 + 6).
Similarly, the area of the skylight is given as (𝑥 + 5) * (3𝑥 + 4).
Now, let's calculate the area of the remaining roof:
Area of remaining roof = Area of original roof - Area of skylight
= (10𝑥 + 8) * (8𝑥 + 6) - (𝑥 + 5) * (3𝑥 + 4)
Expanding both expressions, we get:
Area of remaining roof = (80𝑥^2 + 60 + 64𝑥 + 48) - (3𝑥^2 + 4𝑥 + 15𝑥 + 20)
= 80𝑥^2 + 64𝑥 + 60 + 48 - 3𝑥^2 - 4𝑥 - 15𝑥 - 20
= 77𝑥^2 + 45𝑥 + 88
Therefore, the area of the remaining roof after the skylight is built is given by the expression 77𝑥^2 + 45𝑥 + 88.
Hence, the correct answer is not listed among the answer choices provided.