triangle base is (4-x)cm and its height is (2x+3)cm.what will be the maximium area?
Area = (1/2)base x heigh
= (1/2)(4-x)(2x+3)
= (1/2)(12 + 5x - 2x^2)
d(Area)/dx = (1/2)(5 - 4x)
= 0 for a max area
4x = 5
x = 5/4
using x = 5/4 = 1/25
Maximum Area = (1/2)(4-5/4)(5/2+3) = 7.5625
or appr 7.5625
testing: our answer t= 5/4 = 1.25
let x = 1.24 , area = (1/2)(4-1.24)(2.48+3) = 7.5624 , a bit smaller
let x = 1.26, area = (1/2)(4-1.26)(2.52+3) = 7.5624 , a bit smaller again
all looks good at max area = 7.5625
can you explain how you got 1/2 x 5-4x ?
Since this is a typical Calculus question, I assume you knew Calculus.
I took the derivative.
You could start with my area equation
area = (1/2)(12 + 5x - 2x^2)
= (1/2)(-2)(x^2 - 5/2x -6)
= -(x^2 - (5/2)x + 25/16 - 25/16 - 6) ----I competed the square
= -( (x - 5/4)^2 - 25/16 - 6)
= (x - 5/4)^2 + 25/16 + 6
= -(x-5/4)^2 + 121/16
area = -(x-5/4)^2 + 121/16
max area = 121/16 or 7.5625
when x = 5/4
just as in my first solution.
To find the maximum area of a triangle with a given base and height, you need to use calculus. The formula to calculate the area of a triangle is given by:
Area = (1/2) * base * height
In this case, the base of the triangle is given as (4 - x) cm and the height is given as (2x + 3) cm.
Therefore, the area of the triangle can be expressed as:
Area = (1/2) * (4 - x) * (2x + 3)
To find the maximum area, you need to find the critical points of this equation. Critical points are the values of x where the derivative of the equation is equal to zero or is undefined. So, let's find the derivative first:
Area' = d(Area) / dx = (1/2) * ((-1) * (2x + 3) + (4 - x) * (2)) = (1/2) * (-2x - 3 - 2x + 8)
= (1/2) * (-4x + 5)
Now, set the derivative equal to zero and solve for x:
(1/2) * (-4x + 5) = 0
-4x + 5 = 0
-4x = -5
x = 5/4
Now, we have found a critical point at x = 5/4. To confirm if it is a maximum or minimum, we can take the second derivative:
Area'' = d²(Area) / dx² = (1/2) * (-4)
Since the second derivative is negative (-2), this means that the function is concave down, which indicates a maximum.
Therefore, the maximum area of the triangle can be found by substituting x = 5/4 into the equation:
Area = (1/2) * (4 - 5/4) * (2(5/4) + 3)
= (1/2) * (16/4 - 5/4) * (10/4 + 12/4)
= (1/2) * (11/4) * (22/4)
= (11/8) * (22/4)
= 55/8
So, the maximum area of the triangle is 55/8 square centimeters.