Im stuck on this question i need help

Mathematics

1 answer:

Korolek [52]3 years ago

4 0

I think it’s one half 1/2

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One side of a triangle is increasing at a rate of 9 cm/s and a second side is decreasing at a rate of 2 cm/s. if the area of the

LenKa [72]

Use the attached figure to guide yourself through the explanation. In the figure:
$a=21cm, b=36cm, \widehat{C}=\frac{\pi}{6}$ .
We know that:
$\frac{da}{dt}=9,\frac{db}{dt}=-2$
Also remmeber that the area of a triangle is:
$A=\frac{Base\times Height}{2}$
The hegiht is shown in the figure in red, the base is just the b side.
Using the angle C the area of the triangle is:
$A=\frac{1}{2}ab\cos(\widehat{C})\implies2A=ab\cos(\widehat{C})$
Now taking the derivative of the area 2A with respect to t we have:
$2\frac{dA}{dt}=b\cos(\widehat{C})\frac{da}{dt}+a\cos(\widehat{C})\frac{db}{dt}-ab\sin(\widehat{C})\frac{d\widehat{C}}{dt}$

Replacing with the right values:
$\frac{dA}{dt}=0\\
\\
\cos({\widehat{C}})=\frac{\sqrt{3}}{2}\\ \\
\sin({\widehat{C}})=\frac{1}{2}\\ \\

$
yields:
$0=36\frac{\sqrt{3}}{2}.9-21\frac{\sqrt{3}}{2}.2-\frac{1}{2}.21.36.\frac{d\widehat{C}}{dt}\implies \frac{d\widehat{C}}{dt}=\frac{141}{378}\sqrt{3}$