An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna. When the plane is 10 miles away (s = 10), the radar detects that the distances is changing at a rate of 240 miles per hour. What is the speed of the plane?

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Upout1940 · Accepted Answer

We are given the Distance s of the plane, the Height h and the rate of change of Distance at the given distance&amp;nbsp;ds&amp;nbsp;dt&amp;nbsp;&amp;nbsp;as follows:&amp;nbsp;s=10&amp;nbsp;mi&amp;nbsp;h=5&amp;nbsp;mi&amp;nbsp;ds&amp;nbsp;dt&amp;nbsp;∣s=10=240mih&amp;nbsp;We can use the Pythagorean Theorem:&amp;nbsp;h2+x2=s2&amp;nbsp;The following, we will want to differentiate both sides with respect to t, but doing so yields.:&amp;nbsp;d&amp;nbsp;dt&amp;nbsp;(h2+x2)=d&amp;nbsp;dt&amp;nbsp;(s2)&amp;nbsp;So now using the chain rule and differentiation tables we get:&amp;nbsp;d&amp;nbsp;dt&amp;nbsp;(h2+x2)=d&amp;nbsp;dt&amp;nbsp;(s2)&amp;nbsp;d&amp;nbsp;dt&amp;nbsp;(h2)+d&amp;nbsp;dt&amp;nbsp;(x2)=d&amp;nbsp;dt&amp;nbsp;(s2)&amp;nbsp;0+&amp;nbsp;dx&amp;nbsp;&amp;nbsp;dx&amp;nbsp;d&amp;nbsp;dt&amp;nbsp;(x2)=dsdsd&amp;nbsp;dt&amp;nbsp;(s2)&amp;nbsp;&amp;nbsp;dx&amp;nbsp;&amp;nbsp;dt&amp;nbsp;d&amp;nbsp;dx&amp;nbsp;(x2)=ds&amp;nbsp;dt&amp;nbsp;dds(s2)&amp;nbsp;&amp;nbsp;dx&amp;nbsp;&amp;nbsp;dt&amp;nbsp;2x=ds&amp;nbsp;dt&amp;nbsp;2s&amp;nbsp;x&amp;nbsp;dx&amp;nbsp;&amp;nbsp;dt&amp;nbsp;=ds&amp;nbsp;dt&amp;nbsp;s&amp;nbsp;What we want to discover is&amp;nbsp;&amp;nbsp;dx&amp;nbsp;&amp;nbsp;dt&amp;nbsp;&amp;nbsp;that is the speed so solving for it:&amp;nbsp;&amp;nbsp;dx&amp;nbsp;&amp;nbsp;dt&amp;nbsp;=sxds&amp;nbsp;dt&amp;nbsp;&amp;nbsp;We dont

An airplane is flying at an altitude of 5 miles and passes directly over a radar

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