Question

the sum of the base and the height of a triangle is 14 cm. find the dimensions for which the area is a maximum. the triangle with maximum area has a height of cm and a base of cm.

Explanation:

Step1: Define variables

Let the base of the triangle be $b$ and the height be $h$. Given $b + h=14$, so $h = 14 - b$. The area formula of a triangle is $A=\frac{1}{2}bh$.

Step2: Substitute $h$ into area formula

Substitute $h = 14 - b$ into $A=\frac{1}{2}bh$, we get $A=\frac{1}{2}b(14 - b)=\frac{1}{2}(14b - b^{2}) = 7b-\frac{1}{2}b^{2}$.

Step3: Find the derivative of the area function

The derivative of $A(b)=7b-\frac{1}{2}b^{2}$ with respect to $b$ is $A'(b)=7 - b$.

Step4: Set the derivative equal to zero

Set $A'(b)=0$, so $7 - b = 0$, which gives $b = 7$.

Step5: Find the second - derivative

The second - derivative $A''(b)=-1<0$. So when $b = 7$, the area function has a maximum.

Step6: Find the height

Since $h = 14 - b$ and $b = 7$, then $h=14 - 7=7$.

Answer:

The triangle with maximum area has a height of $7$ cm and a base of $7$ cm.