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3 years ago

15

Lionel plays trumpet for a minimum of 45 minutes on the days that he practices. If x is the number of days that Lionel practices

and y is the total number of hours he spends practicing, which inequality represents this situation?
y≥0.75x
0.75y≥x
y≤0.75x
0.75y≤x

1 answer:

Lemur [1.5K]3 years ago

4 0

Answer:

y<u>></u>0.75x

Step-by-step explanation:

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Answer the question in the picture

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Recall the angle sum identities:

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$\cos(x+y)=\cos x\cos y-\sin x\sin y$

Now,

$\tan(x+y)=\dfrac{\sin(x+y)}{\cos(x+y)}=\dfrac{\sin x\cos y+\cos x\sin y}{\cos x\cos y-\sin x\sin y}$

Divide through numerator and denominator by $\cos x\cos y$ to get

$\tan(x+y)=\dfrac{\tan x+\tan y}{1-\tan x\tan y}$

Next, we use the fact that $x,y$ lie in the first quadrant to determine that

$\sin x=\dfrac12\implies\cos x=\sqrt{1-\sin^2x}=\dfrac{\sqrt3}2$

$\cos y=\dfrac{\sqrt2}2\implies\sin x=\sqrt{1-\cos^2x}=\dfrac1{\sqrt2}$

So we then have

$\tan x=\dfrac{\sin x}{\cos x}=\dfrac{\frac12}{\frac{\sqrt3}2}=\dfrac1{\sqrt3}$

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