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lys-0071 [83]
2 years ago
8

Y varies directly as x, if y=9 x=3 find x when y=-27

Mathematics
1 answer:
jonny [76]2 years ago
4 0

Answer:

x = -9

Step-by-step explanation:

Standard equation of direct variation:

y = kx

We need to find k.

We use the given point.

When x = 3, y = 9.

9 = k * 3

k = 3

Our equation is

y = 3x

Now we find x when y = -27.

y = 3x

-27 = 3x

x = -9

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Step-by-step explanation:

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Express the function y(t)= 4 sin 2πt + 15 cos 2πt in terms of (a) a sine term only. (b) Determine the amplitude, the period, the
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Answer:

See the step by step solution

Step-by-step explanation:

Consider a more general case of the form a\sin(x)+b\cos(x). We want to express it as a sine term only by using the following formula \sin(\alpha+\beta) = \sin(\alpha)\cos(\beta)+\sin(\beta)\cos(\alpha)

The trick  consists on find a number \beta \in [0,2\pi] such that \cos(\beta) = a, \sin(\beta)=b . But, note that since \cos^2(\beta)+\sen^2(\beta)=1 it is most likely that a^2+b^2neq 1. Then, we will use the following equations:

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Then, problem turns out to be

a\sin(x)+b\cos(x)= \sqrt[]{a^2+b^2}(A\sin(x)+B\cos(x)) =\sqrt[]{a^2+b^2}(\sin(x)\cos(\beta)+\sin(\beta)\cos(x))=\sqrt[]{a^2+b^2}\sin(x+\beta),

where \beta = \tan^{-1}(\frac{B}{A})= \tan^{-1}(\frac{b}{a}). So, note that the frequency is the same. Then, a) Taking a=4 and b= 15 we have that 4 sin 2πt + 15 cos 2πt= [tex]\sqrt[]{15^2+4^2}\sin(2\pi t + \tan^{-1}\frac{15}{4})

b) The amplitude is \sqrt[]{15^2+4^2} = \sqrt[]{241}. Since 2\pi rad/s = 1 Hz the frequency is 1 Hz. The period is given by \frac{2\pi}{2\pi} = 1.

c) The function's graph is attached

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