The length of leg a of a right triangle is increasing at a rate of 2 cm per minu

Question

The length of leg a of a right triangle is increasing at a rate of 2 cm per minute and the length of the hypoteneuse c is increasing at a rate of 5 cm per minute. At what rate is the length of the other leg b increasing when leg a is 6 cm long and c is 10 cm long? Solution Since the derivative is a rate of change (section 3.6). we are given that da/dt = 2 and dc/dt = 5. We want to find out what db/dt is when a = 6 and c = 10. By the Pythagorean Theorem, a^2 + b^2 = c^2 and so implicitly differentiating gives that 2a da/dt + 2b db/dt = 2c dc/dt. Since da/dt = 2 and dc/dt = 5, we then have that 2a(2) + 2b db/dt = 2c(5) so that 2a + b db/dt = 5c. When a = 6 and c = 10. the Pythagorean Theorem gives that 36 + b^2 = 100 so that b = 8. Therefore, when a = 6 and c = 10. from 2a + b db/dt = 5c, we have that 2(6) + 8 db/dt = 5 (10) so the rate at which leg 6 Ls increasing is db/dt = 19/4 cm per minute. Is this solution correct or incorrect? If it is correct, justify each step with the relevant property and if it is incorrect, specify where the incorrect step is and fix it, if possible.

Explanation / Answer

The above solution is correct.

The lenght of leg a is given and it it increasing at a rate 2cm/min. Rate of increse in leg a is denoted by da/dt

The increasing in hypoteneuse is denoted by dc/dt

The increasing in another leg b is denoted by db/dt

According to the Pythagorean theorem it can be written as c^2 = a^2 + b^2. Then calculate the rate of change and substitute the given values we get the final answer

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