the perimeter of a rectangle is 34 feet and it's area is 30 square feet. Find the length and width of the rectangle.

Question

I need to see the steps to solve  the problem.

Answer 1

Brennan,

Step 1: Define perimeter and area in the context of a rectangle in equation form.

 

Perimeter of a rectangle: side 1 + side 2 + side 3 + side 4 = Perimeter. (likely you know this already) Side 1 = side 3 and side 2 = side 4 so the equation can be simplified to: short side (S)*2 + long side (L)*2 = Perimeter (P).

 

Area of a rectangle: base (S) * height (L) = Area (A).

 

In summary: P = S*2+L*2 and A = S*L

 

Step 2: Fill in the equations with numbers from the question and see if we can find the other variables using algebra.

 

P = 34 ft = S*2 + L*2

A = 30 ft^2 = S*L

 

We have two variables we must solve for but thankfully we also have two equations. 

 

Step 3: Solve for the unknowns.

 

34 = 2S + 2L

30 = S*L

 

(Note: I have dropped the units for simplicity in working out the equations, don't forget them though)

I will use substitution.

 

L = 30/S

34 = 2S + 2*(30/S) = 2S + 60/S Multiply the S through and rearrange.

0 = 2S^2 - 34S + 60

0 = S^2 - 17S + 30 Two possible answers are found. 

S = (2, 15) (Note: not an ordered pair) 

 

30 = (2, 15)*L

L = (15, 2)

L = 15 ft

S = 2 ft (just because 15 > 2).

 

 

 

Answer 2

p = 2(length + width)

34 = 2(length + width)

17 = length + width

17 - width (call it w) = length

 

A = length(width)

30 = (17 - w)(w)

30 = 17w - w²

w² - 17w + 30 = 0                  FOIL

(w - 2)(w - 15) = 0

 

The width is 2 and the length is 15.

 

Check:

2(2 + 15)

2(17)

34

 

(2)(15)

30

Answer 3

This math problem can be written as following system of equations:
2(L+W)=34
LW=30

where L and W stands for Length and Width correspondingly. Furthermore,

(L+W)=17
LW=30

 

Substitute: W=17-L and write the following:
L(17-L)=30

 

Next, expand:
17L-L² =30

 

or in canonical form:
L²-17L+30 =0

Solving this quadratic equation:
L=17, W=2 (it could be also L=2 and W=17)

 

(Note: you can use the online quadratic equation solver http://webinfocentral.com/math/QuadraticEquations.aspx)

 

Best regards,