Answer:
- (a) area = 3y^2 +14y +16
- (b) perimeter = 8y +20
Step-by-step explanation:
Part of the point of Algebra is that you can work with things that stand for numbers (almost) as easily as working with numbers themselves. Here, you are given values that stand for the length and width of the rectangle.
(a) You know the area of a rectangle is found by multiplying its length by its width. The same is true when length and width are algebraic expressions that stand for those numbers.
<u>Given:</u> length = 3y+8, width = y+2
<u>Find:</u> area = length × width
<u>Solution:</u>
area = (3y+8) × (y+2)
Using the distributive property, we can perform this multiplication in essentially the same way we would for the numbers 38 and 12.
area = 3y(y +2) +8(y +2) equivalent to 30(12) +8(12)
= 3y^2 +6y +8y +16 equivalent to 30·10 +30·2 +8·10 +8·2
area = 3y^2 +14y +16 equivalent to 3·10^2 +14·10 +16
(Of course, with numbers, there are relationships between units and tens and hundreds that let us further combine terms. When variables are involved, we can combine terms with the same power of the variable, but we cannot combine "y" terms with "y^2" terms the way we can with numbers.)
___
(b) The perimeter is the sum of the lengths of the four sides of the rectangle. Since two of those sides are "length" and two are "width", we can write ...
perimeter = 2(length + width)
Making the substitution for length and width as defined above, we have
perimeter = 2((3y+8) + (y+2))
= 2(3y+y + 8+2) = 2(4y +10)
perimeter = 8y +20
