the length of the rectangle is 7ft more than twice the width, and the area of the rectangle is 99ft squared. find length and width

Question

Find the dimensions of the rectangle 

Answer 1

Hi Justice,

 

If L is the length and W is the width then L = 7 + 2W.

 

We know that area of a rectangle is given by A= LW. By substituting L in terms of W we can rewrite the area equation as A= (7+2W) W which is given to be 99 sq.ft.

 

So 99 = 2W² + 7W. You can solve the quadratic by using either the quadratic formula or factoring. We get two solutions for W. One is 5.5 and the other is -9. Since the width cannot be a negative value therefore we discard that solution and use W = 5.5 and solve for L which gives us 18.

 

You can double-check the answer by multiplying L and W to make sure that the Area is 99 sq.ft. 5.5ft x 18ft = 99 sq.ft which checks out.

 

Hope it makes sense. Please feel free to message me if you need more clarification.

 

Krish.

Answer 2

Let the width of the rectangle be  x. 

Twice the width is 2x and 7 more would be +7. Therefore the length is 2x + 7 

 

The area of the rectangle is A= w•l . Using the formula we can write our equation  x(2x + 7) = 99. 

 

Applying distributive property we get 2x² + 7x = 99

 

Subtracting  99 from both sides will give us  2x² + 7x - 99 = 0 

 

Factoring the quadratic equation will give us (2x - 11)(x + 9 ) = 0

 

Solve: 2x-11 = 0 , 2x = 11, x = 5.5 

x+9 = 0, x= -9 - since width cannot be negative, -9 is not a solution. 

 

Therefore, the width is 5.5 ft.

 

To find the length we will substitute 5.5 into 2x+7. 

2•5.5 +7 = 18. Thus, the length is 18 ft. 

 

Check the solution: 99 = 5.5 • 18

                               99 = 99

 

Answer: width is 5.5 ft, length is 18 ft. 

 

Answer 3

Let l = length
Let w = width

l = 2w + 7

Let A = area

A = l * w

Use substitution by using the formula for l in the formula for A:

A = l * w
A = (2w + 7) * w
A = 2w2 + 7w

We know that A = 99, so:

99 = 2w2 + 7w

 

Now we want to set it equal to zero so we can solve for w:

2w2 + 7w - 99 = 0

Solve using the quadratic formula:

w = (-b +/- √(b2 - 4ac)) / 2a
where a = 2, b = 7, and c = -99

 

Solve for the value under the radical first:

b2 - 4ac
72 - 4(2)(-99) = 841

 

Then solve for the numerator:

-b +/- √(b² - 4ac)

-7 +/- √841

-7 +/- 29

 

Now solve for the denominator

 

2a = 2(2) = 4

 

Since there is a plus-or-minus there, we have 2 possible values for the numerator:

 

-7 + 29 = 22

-7 - 29 = -36

 

Divide both of them by 4:

 

22/4 = 5.5

-36/4 = -9

 

We cannot have a negative value for a distance, so the width must be 5.5. Now to find the length, we simply plug 5.5 back into the very first equation: l = 2w + 7

 

l = 2w + 7

l = 2(5.5) + 7

l = 11 + 7

l = 18

 

So the width is 5.5ft, and the length is 18ft. We can double check this by multiplying 18 by 5.5.

 

18 * 5.5 = 99

 

This is a valid statement, so our width and length are correct.

Answer 4

Hi Justice,

 

The given information concerns the area and the lengths of the sides of the rectangle.The formula that connects these pieces of information is the formula for the area of a rectangle. Let x represent the width of the rectangle. Then the length will be 2x + 7.

                                   A = length times the width

 

                                  99 = x(2x + 7)

 

                                  99 = 2x² + 7x  Write the equation in standard form as 

 

                                   0  = 2x² +7x - 99    Factor the right hand side of the equation to get

 

                                   0 = (2x - 11)(x + 9)    Set each factor equal to zero and solve

 

                                      2x - 11 = 0      or      x + 9 = 0

 

                                              x = 11/2                x = -9 A side can not have a negative measure so discard -9

We must find the other side represented by 2x + 7  or  2(11/2) + 7 = 18   

 

Therefore the dimensions of the rectangle are  5.5 by 18.