What is \angle A equal to if the area of the triangle is <strong>MAXIMUM</strong>?
Consider the following for the two (02) items that follow : Let ABC be a triangle right-angled at B and AB+AC=3 units.
- A. \pi/6
- B. \pi/4
- C. \pi/3 ✓
- D. 5\pi/12
Correct Answer: C. \pi/3
Explanation
Let AB = c, AC = b, BC = a. Given c + b = 3 \implies b = 3 - c. By Pythagoras theorem, a^2 + c^2 = b^2 = (3-c)^2 = 9 - 6c + c^2 \implies a^2 = 9 - 6c. The area is \Delta = \frac{1}{2}ac. To maximize \Delta, we maximize \Delta^2 = \frac{1}{4}c^2(9-6c) = \frac{1}{4}(9c^2 - 6c^3). Differentiating wrt c and equating to 0: 18c - 18c^2 = 0 \implies c = 1. Then b = 3 - 1 = 2. Therefore, \cos A = \frac{c}{b} = \frac{1}{2} \implies \angle A = \frac{\pi}{3}.
Related questions on Calculus
- Let z=[y] and y=[x]-x, where [.] is the greatest integer function. If x is <strong>NOT</strong> an integer but positive, then what i...
- If f(x)=4x+1 and g(x)=kx+2 such that fog(x)=gof(x), then what is the value of k?
- What is the <strong>MINIMUM</strong> value of the function f(x)=\log_{10}(x^{2}+2x+11)?
- What is \int(x^{x})^{2}(1+\ln x)\,dx equal to ?
- What is \int e^{x}\{1+\ln x+x\ln x\}\,dx equal to?