triangle base is (4-x)cm and its height is (2x+3)cm.what will be the maximium area?
To find the maximum area of a triangle, we can use calculus.
The area of a triangle is given by the formula: A = (1/2) * base * height.
In this case, the base of the triangle is (4-x) cm and the height is (2x+3) cm.
So, the area A can be written as: A = (1/2) * (4-x) * (2x+3).
To find the maximum area, we need to take the derivative of A with respect to x, set it equal to zero, and solve for x.
Let's do that:
First, expand the equation: A = (1/2) * (8x - 2x^2 + 12 - 3x).
Now, differentiate A with respect to x: A' = 8/2 - 2(2x) + 0 - 3/2.
Simplifying: A' = 4 - 4x - 3/2.
Now, set A' equal to zero and solve for x:
4 - 4x - 3/2 = 0.
To get rid of the fraction, multiply the equation by 2: 8 - 8x - 3 = 0.
Rearrange the equation: -8x = -5.
Finally, solve for x: x = 5/8.
To find the maximum area, substitute this value of x back into the original equation for A:
A = (1/2) * (4 - 5/8) * (2(5/8) + 3).
Simplifying the expression, A = (1/2) * (27/8) * (31/8).
A = 837/64 cm^2.
Therefore, the maximum area of the triangle is 837/64 cm^2.