Answer:
width = 10 inches; length = 14 inches
Explanation:
- It appears that you're trying to find the dimensions of the poster given the info about the length and the width.
- If this is the case, my work below will help you find these dimensions.
- If you're trying to find something else, write it in the comments and I'll edit my answer to help you.
The formula for the area of a rectangle:
The formula for the area of a rectangle is given by:
A = lw, where
- A is the area in square units,
- l is the length,
- and w is the width
Relating length and width:
Since we're told that the poster's length is 9 more inches than half its width, we can model this with the following equation:
l = 1/2w + 9
Finding the width:
Now we can substitute 140 for A and 1/2w + 9 for w to find w, the width of the poster:
140 = (1/2w + 9)w
140 = 1/2w^2 + 9w
Putting the 140 = 1/2w^2 + 9w in standard form (ax^2 + bx + c = 0)
We can subtract 1/2w^2 and 9w from both sides to put the quadratic in standard form, which will allow us to solve it:
(140 = 1/2w^2 + 9w) - 1/2w^2 - 9w
-1/2w^2 - 9w + 140 = 0
Clear the fraction in -1/2w^2 to simplify the quadratic:
In order to clear the -1/2 from -1/2w^2, we can divide both sides both sides by -1/2 (note that when dividing by a fraction, we multiply by the reciprocal, so we'll end up both sides multiplying by -2:
(-1/2w^2 - 9w + 140 = 0) / -1/2
(-1/2w^2 - 9w + 140 = 0) * -2
1w^2 + 18w - 280 = 0
w^2 + 18w - 280 = 0
Thus, our a value is 1, our b value is 18, and our c value is -280.
Solving w^2 + 18w - 280 = 0 using the quadratic formula:
Now we can solve the quadratic using the quadratic formula, which is given by:
x = -b / 2a ±( √(b^2 - 4ac)) / 2a, where
- x is a solution to the quadratic,
Since we're solving for w, we can swap x with w in the quadratic equation to prevent confusion:
Thus, we can plug in 1 for a, 18 for b, and -280
w = -18 / 2(1) ± (√((18)^2 - 4(1)(-280))) / 2(1)
w = -18 / 2 ± (√1444) / 2
w = -9 ± 38/2
w = -9 + 19 and x = -9 - 19
w = 10 and x = -28
Thus, w = 10 and w = -28.
Since we can't have a negative dimension, the width is 10 inches.
Determine the length:
Now we can find the poster's length by plugging in 140 for A and 10 for w in the rectangle area formula and solving for l:
(140 = 10l) / 10
14 = l
Thus, the poster's length is 14 inches.
Check the validity of the answer:
We can check that we've correctly determined the poster's dimensions by seeing if the product of 10 and 14 is 140 and if half of 10 plus 9 = 14.
10 * 14 = 140
1/2 * 10 = 5
5 + 9 = 14
Thus, our answers are correct.