The area of a rectangle is 220 square centimeters. The height of the rectangle is two more than twice the base. What is the rectangle's perimeter?
I know you can solve this employing a guess-and-check strategy, but I want to know if this can be solved algebraically.
So far, I've performed every substitution under the sun; I keep arriving at contradictory deductions.
let the base be b, let the height be h
then it says.. h = 2b + 2
area of rectangle is base*height
b(2b+2)= 220
b^2 + 2b - 220 = 0
Do you know how to solve a quadratic?
my equation should have said
2b^2 + 2b - 220 = 0
it comes out to an integer value of b
Eureka. Sometimes we can't see the forest for the trees. Thank you immensely.
Yes, this problem can indeed be solved algebraically using the equation you've derived.
To find the perimeter of the rectangle, we need to know both the base and the height. We already have an equation relating the height (h) to the base (b), which is h = 2b + 2.
The area of the rectangle is given as 220 square centimeters, so we can set up an equation using the formula for the area of a rectangle: A = base * height. Plugging in the values, we get:
b * (2b + 2) = 220
Simplifying this equation, we have:
2b^2 + 2b - 220 = 0
Now, to solve this quadratic equation, we can use various methods, such as factoring, completing the square, or using the quadratic formula. In this case, factoring might not be efficient, so let's use the quadratic formula:
b = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values from our equation, we have:
b = (-(2) ± √((2)^2 - 4(2)(-220))) / (2(2))
Simplifying further:
b = (-2 ± √(4 + 1760)) / 4
b = (-2 ± √1764) / 4
b = (-2 ± 42) / 4
Since we are looking for a positive value for the base, we can discard the negative solution:
b = (-2 + 42) / 4
b = 40 / 4
b = 10
So the base of the rectangle is 10 centimeters.
Now, we can substitute this value back into the equation for the height:
h = 2b + 2
h = 2(10) + 2
h = 20 + 2
h = 22
Therefore, the height of the rectangle is 22 centimeters.
To find the perimeter, we add up all the sides:
Perimeter = 2(base) + 2(height)
Perimeter = 2(10) + 2(22)
Perimeter = 20 + 44
Perimeter = 64
So the perimeter of the rectangle is 64 centimeters.