a triangluar garden has an area of 189 square feet. its height os 3 feet more than ots base. find the measure of the base.
base ---- x
height ---- x+3
(1/2)x(x+3) = 189
x^2 + 3x = 378
x^2 + 3x - 378 = 0
(x-18)(x+21) = 0
x=18 or x=-21, but x can't be negative, so
x=18
base is 18, height is 21
check:area = (1/2)(18x21) = 189
A = Area
b = base
h = height
h = b + 3
A = b * h / 2 = 189 ft ^ 2
b * h / 2 = b * ( b + 3 ) / 2 = 189 Multiply both sides by 2
b * ( b + 3 ) = 378
b ^ 2 + 3 b - 378 = 0
The solutions are :
b = 18 ft
and
b = - 21 ft
The base can't be negative,so solution are :
b = 18 ft
h = b + 3 = 21 ft
Proof :
A = b * h / 2 = ( 18 * 21 ) / 2 = 378 / 2 = 189 ft ^ 2
If you don't know how to solve equaion
b ^ 2 + 3 b - 378 = 0
In google type:
quadratic equation online
When you see list of results click on:
Free Online Quadratic Equation Solver:Solve by Quadratic Formula
When page be open in rectangle type:
b ^ 2 + 3 b - 378 = 0
and click option: solve it!
You wil see solution step-by-step
To find the measure of the base of the triangular garden, we can set up an equation using the formula for the area of a triangle:
Area = (base * height) / 2
Given that the area is 189 square feet and the height is 3 feet more than the base, we can represent the equation as:
189 = (base * (base + 3)) / 2
To solve this equation for the base, we can rearrange it and solve for the base:
Multiply both sides of the equation by 2:
2 * 189 = base * (base + 3)
Simplify the equation:
378 = base^2 + 3base
Rearrange the equation:
base^2 + 3base - 378 = 0
Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
base = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = 3, and c = -378. Substituting these values in the quadratic formula:
base = (-3 ± √(3^2 - 4 * 1 * -378)) / (2 * 1)
Simplify the equation under the square root:
base = (-3 ± √(9 + 1512)) / 2
base = (-3 ± √1521) / 2
base = (-3 ± 39) / 2
Now, we have two possible solutions for the base: (-3 + 39)/2 and (-3 - 39)/2.
Calculating the first solution:
base = (-3 + 39)/2
base = 36/2
base = 18
Calculating the second solution:
base = (-3 - 39)/2
base = -42/2
base = -21
Since the base of a garden cannot be negative, the measure of the base is 18 feet.