document.write( "Question 1119873: The base and the base angles of an isosceles triangle are increasing at the respective rates of 2 ft/s and 5 degrees/sec. When the base is 10 ft long and the base angles are 45°, find the rate at which the altitude is increasing? \n" ); document.write( "

Algebra.Com's Answer #735530 by greenestamps(13195) $\"\"$ $\"About$

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\n" ); document.write( "Let x be the measure of the base angle, b be the length of the base, and h be the height. Then

\n" ); document.write( " $\"tan%28x%29+=+h%2F%28b%2F2%29+=+2h%2Fb\"$
\n" ); document.write( " $\"2h+=+b%2Atan%28x%29\"$

\n" ); document.write( "Take the derivative with respect to time, using the product rule on the right:

\n" ); document.write( " $\"2%2A%28dh%2Fdt%29+=+%28db%2Fdt%29%2Atan%28x%29%2Bb%2A%28sec%28x%29%5E2%29%28dx%2Fdt%29\"$

\n" ); document.write( "Plug in the given rates of change of the base and the base angle, noting that the rate of change of the angle must be in radians per second.

\n" ); document.write( " $\"2%2A%28dh%2Fdt%29+=+%282%29%2Atan%28x%29%2Bb%2A%28sec%28x%29%5E2%29%28%28pi%29%2F36%29\"$

\n" ); document.write( "And evaluate when the base is 10 and the angle x is 45 degrees (tan(x) = 1; sec(x)^2 = 2):

\n" ); document.write( " $\"2%2A%28dh%2Fdt%29+=+%282%29%281%29%2B%2810%29%282%29%28%28pi%29%2F36%29\"$
\n" ); document.write( " $\"2%2A%28dh%2Fdt%29+=+2+%2B+%285%2F9%29%28pi%29%29\"$
\n" ); document.write( " $\"dh%2Fdt+=+1+%2B+%285%2F18%29%28pi%29\"$ \n" ); document.write( "

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