Calculus

Application of Derivatives

Medium

core

The sides of an equilateral triangle are increasing at the rate of 5 cm/sec. The rate at which the area increases when the side is 20 cm, is

$25\sqrt{3}$ cm²/sec

$50\sqrt{3}$ cm²/sec

$100\sqrt{3}$ cm²/sec

$15\sqrt{3}$ cm²/sec

✅ Correct Option: 2

The sides of an equilateral triangle are increasing at the rate of 5 cm/sec, with the current side length at 20 cm.

The area of an equilateral triangle with side $a$ is given by:

$A = \dfrac{\sqrt{3}}{4} \times a^2$

Differentiating both sides with respect to time $t$ :

$\dfrac{dA}{dt} = \dfrac{\sqrt{3}}{4} \times \dfrac{d(a^2)}{dt}$

$\dfrac{dA}{dt} = \dfrac{\sqrt{3}}{4} \times 2a \times \dfrac{da}{dt}$

$\dfrac{dA}{dt} = \dfrac{\sqrt{3}}{2} \times a \times \dfrac{da}{dt}$

Given:

$a = 20$ cm

$\dfrac{da}{dt} = 5$ cm/sec

Substituting the values:

$\dfrac{dA}{dt} = \dfrac{\sqrt{3}}{2} \times 20 \times 5$

$\dfrac{dA}{dt} = \dfrac{\sqrt{3}}{2} \times 100$

$\dfrac{dA}{dt} = 50\sqrt{3}$ cm²/sec

Therefore, the rate at which the area increases is $50\sqrt{3}$ cm²/sec.

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For the function $f(x)=2x^3-9x^2+12x-5, x \in [0,3]$ , which of the following statements are correct ?

A. The point of absolute maxima is x=3.

B. The absolute minimum value is -1.

C. The critical points are x=1 and x=2.

D. The absolute maximum value is 3.

Choose the correct answer from the options given below:

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Based on above information answer the following question :

$x$ and $y$ will satisfy :

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The sides of an equilateral triangle are increasing at the rate of 5 cm/sec. The rate at which the area increases when the side is 20 cm, is

Solution

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