Question #315591

The Altitude of a triangle is increasing at a rate of 8cm/s while its area is increasing at the rate of 12cm^2/s. At what rate is the base of the triangle changing when the altitude is 20 cm and the area is 100 cm^2 ?


1
Expert's answer
2022-03-22T19:21:11-0400

Given;

dadt=8cm/mindAdt=12cm2/min\\\frac{da}{dt}=8cm/min\\\frac{dA}{dt}={12}cm^2/min with a=altitude, A=area, t=time

The question requires us to find dbdt\\\frac{db}{dt} when a=20cm and A=100cm2.

A=12baddt(A)=ddt(12ba)dAdt=dbdta+12bdadtA=\frac{1}{2}ba\\\frac{d}{dt}(A)=\frac{d}{dt}(\frac{1}{2}ba)\\\frac{dA}{dt}=\frac{db}{dt}a+\frac{1}{2}b\frac{da}{dt}

Using the formula for area, we can tell that b=10cm.

Substituting ;

12=12dbdt20+12(10)(8)dbdt=2.8cm/min12=\frac{1}{2}\frac{db}{dt}20+\frac{1}{2}(10)(8)\\\frac{db}{dt}=-2.8cm/min

The base is changing at a rate of -2.8cm/min




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