A projector displays an image on a wall. The area (in square feet) of the projection is represented by $x^2 - 8x + 15$. The width is $(x-3)$ ft and the height is $x$ ft a. Write a... A projector displays an image on a wall. The area (in square feet) of the projection is represented by $x^2 - 8x + 15$. The width is $(x-3)$ ft and the height is $x$ ft a. Write a binomial that represents the height of the projection. b. Find the perimeter of the projection when the height of the wall is 8 feet.
Understand the Problem
The question describes a projection on a wall with a given area represented by a quadratic expression. Part (a) asks to find a binomial representing the height of the projection. Part (b) asks to find the perimeter of the projection when the height of the wall is 8 feet.
Answer
a. $x-5$ b. $16$
Answer for screen readers
a. $x-5$ b. $16$
Steps to Solve
- Factor the quadratic expression representing the area
We need to factor the quadratic expression $x^2 - 8x + 15$ to find the binomials that represent the length and height of the projection.
$x^2 - 8x + 15 = (x - 3)(x - 5)$
- Determine the height
Since the width of the projection is given as $(x - 3)$ ft, the height must be the other factor, which is $(x - 5)$ ft.
- Substitute the given height of the wall into the expressions for the length and height
The height of the wall is given as 8 feet, and the height of the projection is represented by $x$. We set $x = 8$. Then the width of the projection is $x - 3 = 8 - 3 = 5$ feet, and the height is $x - 5 = 8 - 5 = 3$ feet.
- Calculate the perimeter
The perimeter of a rectangle is given by the formula $P = 2(\text{length} + \text{width})$. In this case, the length is 5 feet and the height is 3 feet. So the perimeter is:
$P = 2(5 + 3) = 2(8) = 16$ feet.
a. $x-5$ b. $16$
More Information
The problem involves factoring a quadratic expression to find the dimensions of a rectangle. Then, it uses substitution and the formula for the perimeter of a rectangle to arrive at the answer.
Tips
A common mistake is to confuse area and perimeter. Remember that area is the space inside a 2D shape, whereas the perimeter is the distance around the outside of the shape. Another mistake is to incorrectly factor the quadratic equation. Always double check your factors by multiplying them back out to see if you get the original quadratic equation. Lastly students may incorrectly substitute values into the expressions.
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