long equation help?

Question

A rectangular plaque has perimeter 40 cm. If its area is 75cm², what are the measurements of a.) the width and b.) the length?

Answer 1

Let the sides be x and y.
Then Perim = 40 tells us what x+y is;
area = 75 tells us what xy is. I suggest you consider all the pairs of factors of 75 i.e. 1*75, 3*25, 5*15 and see which pair give the correct perimeter.

However I suspect your teacher wants you to solve the simultaneous equations
xy = 75
x + y = .... Can you do that? If not, scroll down.






























































x+y = 20 (half of 40)
Multiply both sides by x:
x^2 + xy = 20x, and we know that xy =75

x^2 - 20x + 75 = 0, and to factorise this you have to find two numbers which ... guess what? ... multiply to give 75 and add to -20, same as when we do it without equations!!

(x - 5)(x - 15) = 0

x = 5 or 15; and so y = 15 or 5.

Length is 15 cm, width is 5 cm

Answer 2

Remember that a rectangle's perimeter is calculated with the following formula:

P = L + W + L + W (since a rectangle has two lengths and two widths)

Therefore, since we're given P, and we can combine like terms for L and W, we get

40 = 2L + 2W

Note that the area of a rectangle is calculated using the following formula:

A = LW

But we're given that A = 75, so

75 = LW

Now, we have two equations and two unknowns:

40 = 2L + 2W
75 = LW

We can reduce the first equation, by dividing everything by 2

L + W = 20
LW = 75

Now, you have to solve this by substitution. Since you're asked for the width first, let's calculate L in terms of W in the first equation.

L + W = 20
L = 20 - W

Now, let's plug L = 20 - W into the second equation.

LW = 75
(20 - W)W = 75
20W - W^2 = 75

Bring everything over to the right hand side, to get

0 = W^2 - 20W + 75

Now we factor.
0 = (W - 5) (W - 15)

Resulting in W = 5 or W = 15. Both are equally acceptable answers.

To solve for L, we have to let W be both solutions, and plug them into one of the two equations. Let's use the first one.

W = 15:
L + W = 20
L + 15 = 20
L = 5

W = 5:
L + W = 20
L + 5 = 20
L = 15

Therefore, L = 15 and W = 5, OR L = 5 and W = 15. Since length and width have no significant meaning to each other, those two solutions can be combined into one (i.e. saying you have apples and oranges is no different from saying you have oranges and apples).

Therefore, L = 15 and W = 5.

Answer 3

width = 5cm
length = 15 cm

15+15+5+5 = 40 cm perimeter
area = 15 x 5 = 75 cm