The question is:
The height h of an equilateral triangle is increasing at a rate of 3cm/min. How fast is the area changing when h is 5 cm? Give the answer to 2 decimal places.
How can I figure this question out?
Thank You!
The question is:
The height h of an equilateral triangle is increasing at a rate of 3cm/min. How fast is the area changing when h is 5 cm? Give the answer to 2 decimal places.
How can I figure this question out?
Thank You!
1. The area of a triangle is calculated by
$\displaystyle a = \dfrac12 \cdot s \cdot h$
2. In an equilateral triangle the sides are equal and the height is calculated by:
$\displaystyle h^2+\left(\frac12 s\right)^2 = s^2~\implies~h^2 = \dfrac34 s^2~\implies~s = \dfrac23 \sqrt{3} \cdot h$
3. Plug in this term instaed of s into the equation of the area:
$\displaystyle a(h)=\dfrac13 \sqrt{3} \cdot h^2$
4. Calculate the first derivation of a to get the speed of change:
$\displaystyle a'(h) = \dfrac23 \sqrt{3} \cdot h$
Now calculate a'(5):
$\displaystyle a'(5) = \dfrac23 \sqrt{3} \cdot 5 ~\approx 5.77 \ square\ units$