# The sum of the base and the height of a triangle is 28 cm. Find the dimensions for which of the...

## Question:

The sum of the base and the height of a triangle is {eq}28\ cm {/eq}. Find the dimensions for which of the area is a maximum.

## Critical Number of a Function:

Basically the maximum and minimum values of a function occur only at critical numbers but a function may or may not have maximum/minimum values at a critical number. So, first we need to understand the critical number to solve this problem:

Critical number: The numbers x where {eq}f'(x) = 0 {/eq} or {eq}f'(x) {/eq} is undefined but {eq}f(x) {/eq} is defined that number x is called the critical number of {eq}f(x) {/eq}.

Given:

The sum of the base and the height of the triangle is 28 cm.

Let {eq}b {/eq} be the base of the triangle and {eq}h {/eq} be the height of the triangle. So we can define {eq}b + h = 28 \Rightarrow b = 28 - h \ \ ........ (1) {/eq}.

As we know the formula for the area of the triangle is:

\begin{align*} \displaystyle A &= \frac{1}{2} \ b \ h \\ A &= \frac{1}{2} \times (28 - h) \ h &\text{(Plugging in the value of base)}\\ A &= 14 \ h - \frac{h^2}{2} \\ \frac{d A}{dh} &= \frac{d}{dh} \left [ 14 \ h - \frac{h^2}{2} \right ] &\text{(Taking the derivative of area with respect to } h \text{)}\\ &= 14 \times 1 - \frac{1}{2} \times 2 h &\text{(Differentiating using the formula } \frac{d}{dx}[x^n] = n x^{n - 1} \text{)}\\ \frac{d A}{dh} &= 14 - h \\ \frac{d A}{dh} &= 0 &\text{(Plugging in } \frac{d A}{dh} = 0 \text{ for getting the critical numbers)}\\ 14 - h &= 0 \\ - h &= - 14 \\ h &= 14 \ \rm cm &\text{(Multiplying both sides by negative sign)}\\ \end{align*}

Plug in the value of {eq}h {/eq} in the equation (1): {eq}b = 28 - 14 = 14 \ \rm cm {/eq}

Hence the base of the triangle is 14 cm and the height of the triangle is 14 cm.