SOLUTION: area of a rectangle is 243 m2. The height is nine less than 4 times the base. What is the perimeter?

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Question 209727: area of a rectangle is 243 m2. The height is nine less than 4 times the base. What is the perimeter?
Found 2 solutions by drj, MathTherapy:
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
Area of a rectangle is 243 m^2. The height is nine less than 4 times the base. What is the perimeter?

Step 1. Let x be the base and 4x-9 be the height

Step 2. Perimeter P means adding up the four sides of a rectangle.

Step 3. Then P=x%2Bx%2B4x-9%2B4x-9=10x-9

Step 4. Area=x*(4x-9)=243 or 4x^2-9x-243=0 or a quadratic equation.

Step 5. We can use the quadratic formula given as

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=4, b=-9 and c=-243

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B-9x%2B-243+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-9%29%5E2-4%2A4%2A-243=3969.

Discriminant d=3969 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--9%2B-sqrt%28+3969+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-9%29%2Bsqrt%28+3969+%29%29%2F2%5C4+=+9
x%5B2%5D+=+%28-%28-9%29-sqrt%28+3969+%29%29%2F2%5C4+=+-6.75

Quadratic expression 4x%5E2%2B-9x%2B-243 can be factored:
4x%5E2%2B-9x%2B-243+=+4%28x-9%29%2A%28x--6.75%29
Again, the answer is: 9, -6.75. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B-9%2Ax%2B-243+%29



Select the positive number x=9.


Step 6. Then P=x%2Bx%2B4x-9%2B4x-9=10x-9=10%2A9-9=81

Step 7. ANSWER: The perimeter is 81.


I just found this problem on 18 Sep 09, but I hope the above steps were helpful for review. Sorry about the late response.

For free Step-By-Step Videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra or for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.

And good luck in your studies!

Respectfully,
Dr J

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
area of a rectangle is 243 m2. The height is nine less than 4 times the base. What is the perimeter?

Let base of rectangle be B

Since the height is nine less than 4 times the base, then the height is 4B - 9

Since its area = 243m^2, we will have: B(4B - 9) = 243

B(4B - 9) - 243 = 0

4B%5E2+-+9B+-+243+=+0

(4B + 27)(B - 9) = 0

Therefore, 4B = - 27, or, B = 9

We will ignore the negative value as measuremnets CANNOT be negative

This means that B, or base = 9 meters

Now, since base = 9 meters, then height = 4(9) - 9, or, 27

Since base = 9 meters, and height = 27 meters, then perimeter = 2B + 2H ------> 2(9) + 2(27) ------> 18 + 54 = (highlight_green%2872%29 meters.