The area of a rectangle is 90m 2. The base is three less than three times the height. What is the perimeter?
b = 3 h -3
area = b h = (3 h - 3)h = 3 h^2 - 3 h = 90
so
3 h^2 - 3 h - 90 = 0
(h+5)(3h-18) = 0
h = 6 is positive real solution
so
b = 3*6 -3 = 15
then perimeter = 2 h + 2 b
dont get it
how did you get that. not rudely.
Let B = base and H = height
From the first sentence:
B * H = 90
From the second sentence:
B = 3H - 3
Substitute the value of B from the second equation into the first and solve for H. Put the value of H in the second equation to find B. Check by putting both values into the first equation.
As Damon says, "then perimeter = 2 h + 2 b".
I hope this helps. Thanks for asking.
To find the perimeter of a rectangle, we need to know the lengths of both sides, which are the base and the height.
Let's assign variables to the base and the height for easier calculation.
Let's say the base of the rectangle is represented by "b" and the height is represented by "h."
According to the given information, the area of the rectangle is 90m². Since the area of a rectangle is calculated by multiplying the base by the height, we can write the equation:
b * h = 90
Now, it is stated that the base is three less than three times the height. We can write this as an equation:
b = 3h - 3
Now we have two equations representing the given information:
1) b * h = 90
2) b = 3h - 3
We can solve these equations simultaneously to find the values of 'b' and 'h'.
Substituting equation 2) into equation 1), we get:
(3h - 3) * h = 90
Expanding and rearranging the equation, we have:
3h² - 3h - 90 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:
h = (-b ± √(b² - 4ac)) / (2a)
For our quadratic equation, a = 3, b = -3, and c = -90. Substituting these values into the formula, we get:
h = (-(-3) ± √((-3)² - 4 * 3 * -90)) / (2 * 3)
h = (3 ± √(9 + 1080)) / 6
h = (3 ± √1089) / 6
Simplifying the square root, we get two possible values for h:
h₁ = (3 + 33) / 6 = 36 / 6 = 6
h₂ = (3 - 33) / 6 = -30 / 6 = -5
Since the height of a rectangle cannot be negative, we discard h₂ = -5 as an extraneous solution.
Now that we have the height, we can substitute it into equation 2) to find the base:
b = 3h - 3
b = 3(6) - 3
b = 18 - 3
b = 15
So, the height is h = 6m, and the base is b = 15m.
To find the perimeter of the rectangle, we use the formula:
Perimeter = 2 * (base + height)
Substituting the values, we have:
Perimeter = 2 * (15 + 6)
Perimeter = 2 * 21
Perimeter = 42
Therefore, the perimeter of the rectangle is 42 meters.