FActoring Geometry Problem solving!?

Question

Numba 1 The length of a rectangle is 5m greater than twice its width
and its area is 33m squared. Find the dimensions!

Numba 2 The perimeter of a rectanglar piece of property is 8 miles
and its area is 3 square miles Find the dimensions
Hint 1/2P = l + w

Numba 3 Whhen the dimensions of a 2cm x 5cm rectangle were increased by equal amounts the area was increased by 18cm squared Find the dimensiions of the new rectangle

Numba 4 If the sides of a square are increased by 3in , the area becomes 64 in squared Find the lenth of a side of the original square

Numba 5 A rug placed in a 10 ft x 12 ft room covers two-thirds of the floor area and leaves a uniform strip of bare floor around the edges FInd the dimensions of the rug!

numba 6 The area of a rectangulaer pool is 192 square meter. The
lenth of the pool is 4 meters more than its width Find the length and the width!

Answer 1

#1 - If width is x, then length is 2x+5. Area of a rectangle is length times width, so you get:

(2x + 5)(x) = 33
2x^2 + 5x = 33
2x^2 +5x - 33 = 0
(2x +11)(x - 3) = 0

2x + 11 = 0
2x = -11
x = -11/2 (can eliminate, since you can't have a negative measurement)

x - 3 = 0
x = 3

So, width is 3. Then length is 2(3) + 5 = 11.
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#2
P => 2(l+w) = 8
A => lw = 3

2(l+w) = 8
l+w = 4
l = 4-w

lw = 3
(4-w)(w) = 3
4w - w^2 = 3
0 = w^2 -4w +3
0 = (w-3)(w-1)

So, w = 3 or 1

l = 4-w, so if w = 3, l = 1. if w = 1, l = 3.

Either way, the dimensions are 1 and 3.
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#3
current area: 2 x 5 = 10
New rectangle:
(2+x)(5+x) = 18
10 + 7x + x^2 = 18
x^2 + 7x -8 = 0

Now, factor and solve for x.
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#4
(x+3)(x+3) = 64

Solve like the above ones.
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#5
(10-x)(24-x) = (2/3)(10)(24)

solve like above
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#6
width = x
length = 4 + x
(x)(4+x) = 192

solve as above