- #1
lamerali
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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[tex]\theta[/tex] = [tex]\frac{h}{10}[/tex]
[tex]\frac{d\theta}{dt}[/tex] sec[tex]^{2}[/tex] [tex]\theta[/tex] = [tex]\frac{1}{10}[/tex]
[tex]\frac{d\theta}{dt}[/tex] = [tex]\frac{1}{10}[/tex]. cos [tex]^{2}[/tex] [tex]\theta[/tex]
A = [tex]\frac{1}{2}[/tex] b x h
50 = [tex]\frac{1}{2}[/tex] (10) . h
h = 10
tan [tex]\theta[/tex] = [tex]\frac{10}{10}[/tex]
tan [tex]\theta[/tex] = 1
we know,
sin[tex]^{2}[/tex] [tex]\theta[/tex] + cos[tex]^{2}[/tex] [tex]\theta[/tex] = 1
(cos[tex]\theta[/tex] tan [tex]\theta[/tex]) [tex]^{2}[/tex] + cos [tex]^{2}[/tex] [tex]\theta[/tex] = 1
2 cos[tex]^{2}[/tex] [tex]\theta[/tex] = 1
cos[tex]^{2}[/tex] [tex]\theta[/tex] = [tex]\frac{1}{2}[/tex]
[tex]\frac{d\theta}{dt}[/tex] = [tex]\frac{1}{10}[/tex] . cos[tex]^{2}[/tex] [tex]\theta[/tex]
= [tex]\frac{1}{10}[/tex] . cos[tex]^{2}[/tex][tex]\frac{1}{2}[/tex]
= 0.077 rads / min
or 4.6 rads /s
Thanks in advance!
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[tex]\theta[/tex] = [tex]\frac{h}{10}[/tex]
[tex]\frac{d\theta}{dt}[/tex] sec[tex]^{2}[/tex] [tex]\theta[/tex] = [tex]\frac{1}{10}[/tex]
[tex]\frac{d\theta}{dt}[/tex] = [tex]\frac{1}{10}[/tex]. cos [tex]^{2}[/tex] [tex]\theta[/tex]
A = [tex]\frac{1}{2}[/tex] b x h
50 = [tex]\frac{1}{2}[/tex] (10) . h
h = 10
tan [tex]\theta[/tex] = [tex]\frac{10}{10}[/tex]
tan [tex]\theta[/tex] = 1
we know,
sin[tex]^{2}[/tex] [tex]\theta[/tex] + cos[tex]^{2}[/tex] [tex]\theta[/tex] = 1
(cos[tex]\theta[/tex] tan [tex]\theta[/tex]) [tex]^{2}[/tex] + cos [tex]^{2}[/tex] [tex]\theta[/tex] = 1
2 cos[tex]^{2}[/tex] [tex]\theta[/tex] = 1
cos[tex]^{2}[/tex] [tex]\theta[/tex] = [tex]\frac{1}{2}[/tex]
[tex]\frac{d\theta}{dt}[/tex] = [tex]\frac{1}{10}[/tex] . cos[tex]^{2}[/tex] [tex]\theta[/tex]
= [tex]\frac{1}{10}[/tex] . cos[tex]^{2}[/tex][tex]\frac{1}{2}[/tex]
= 0.077 rads / min
or 4.6 rads /s
Thanks in advance!