Derivatives of trig functions and isosceles triangles.

[lamerali]

The base of an isosceles triangle is 20 cm and the altitude is increasing at the rate of 1 cm/min. At what rate is the base angle increasing when the area is 100 cm2?

I wasnt really sure where to start on this question so i tried my best at an answer. I'm sure I've gone wrong with this question so i appreciate any guidance.

the isosceles triangle divides into two right triangles with bases of 10 cm and areas 50 cm2

tan\theta = \frac{h}{10}

\frac{d\theta}{dt} sec^{2} \theta = \frac{1}{10}

\frac{d\theta}{dt} = \frac{1}{10}. cos ^{2} \theta

A = \frac{1}{2} b x h

50 = \frac{1}{2} (10) . h
h = 10

tan \theta = \frac{10}{10}
tan \theta = 1

we know,

sin^{2} \theta + cos^{2} \theta = 1

(cos\theta tan \theta) ^{2} + cos ^{2} \theta = 1

2 cos^{2} \theta = 1
cos^{2} \theta = \frac{1}{2}


\frac{d\theta}{dt} = \frac{1}{10} . cos^{2} \theta

= \frac{1}{10} . cos^{2}\frac{1}{2}
= 0.077 rads / min

or 4.6 rads /s


Thanks in advance!

 

[HallsofIvy]

The base of an isosceles triangle is 20 cm and the altitude is increasing at the rate of 1 cm/min. At what rate is the base angle increasing when the area is 100 cm2?

I wasnt really sure where to start on this question so i tried my best at an answer. I'm sure I've gone wrong with this question so i appreciate any guidance.

the isosceles triangle divides into two right triangles with bases of 10 cm and areas 50 cm2

tan\theta = \frac{h}{10}

\frac{d\theta}{dt} sec^{2} \theta = \frac{1}{10}

\frac{d\theta}{dt} = \frac{1}{10}. cos ^{2} \theta

A = \frac{1}{2} b x h

50 = \frac{1}{2} (10) . h
h = 10

Here's your error. "when the area is 100 cm²" refers to the original isosceles triangle and that had base 20, not 10.

tan \theta = \frac{10}{10}
tan \theta = 1

we know,

sin^{2} \theta + cos^{2} \theta = 1

(cos\theta tan \theta) ^{2} + cos ^{2} \theta = 1

2 cos^{2} \theta = 1
cos^{2} \theta = \frac{1}{2}


\frac{d\theta}{dt} = \frac{1}{10} . cos^{2} \theta

= \frac{1}{10} . cos^{2}\frac{1}{2}
= 0.077 rads / min

or 4.6 rads /s


Thanks in advance!

 

[lamerali]

Here's your error. "when the area is 100 cm²" refers to the original isosceles triangle and that had base 20, not 10.

but if i plug in
A = (1/2) b x h

100 = (1/2) (20) h

h is still equal to 10 cm. Is that the only error you see? because that does not effect the rest of the equation...:S

 

[lamerali]

is the rest of this equation anywhere near correct?? :(