Answer: Let's call the base of the triangle "b" and the height of the triangle "h". We are given that the sum of the base and the height is 22 cm, so we can write:
b + h = 22
We want to find the dimensions for which the area of the triangle is a maximum. The formula for the area of a triangle is:
A = (1/2)bh
We can use the equation b + h = 22 to solve for h in terms of b:
h = 22 - b
We can substitute this expression for h into the formula for the area:
A = (1/2)b(22 - b)
Simplifying this expression, we get:
A = 11b - (1/2)b^2
To find the value of "b" that maximizes the area, we can take the derivative of this expression with respect to "b" and set it equal to zero:
dA/db = 11 - b = 0
Solving for "b", we get:
b = 11
Substituting this value of "b" back into the equation b + h = 22, we get:
h = 22 - b = 22 - 11 = 11
Therefore, the dimensions for which the area of the triangle is a maximum are:
base = 11 cm
height = 11 cm
And the maximum area is:
A = (1/2)bh = (1/2)(11 cm)(11 cm) = 60.5 square cm
Explanation: