Question

A triangle has two sides a = 1cm and b = 2 cm. How fast is the third

side c increasing when the angle α between the given sides is 60°

and is increasing at the rate of 3° per second ?​

Answer 1

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Answer 2

Given:

A triangle has two sides a and b of measure 1cm and 2cm respectively.

Side c increases when α between the given side is 60°

The increasing rate is 3 per second

To Find:

The rate of change of side c

Solution:

It is given that a = 1cm and b = 2cm and measure of angle c is 60°

Now, by using the cosine rule

$\frac{dc}{dt}$ = $\frac{ab sin a}{c} \frac{da}{dt}$

Then using the rule we find the length of the side c

⇒ c = √a² + b² - 2abcosα

       = √1² + 2² - 2×1×2×cos60°

       = √1 + 4 - 2

       = √3

So, c = √3

As we know the quantities of rate of change dc/dt

We substitute the following values:

$\frac{dc}{dt}$ = ab sinα/ c dα/dt

   = 1×2×sin60°/√3 dα/dt

   = 2 √3/2/√3 ×2

Solving the values we get,

= 3

Therefore the third side increases at the rate of 3cm/s².