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3 years ago

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A zoo is building a new large-cat exhibit. Part of the space will be used for lions and part for leopards. The exhibit will hous

e eight large cats in all. Expenses for a lion will be about $8000 per year, and expenses for a leopard will be about $6000 per year. Write an equation that can be used to find y, the yearly expenses for the eight cats in the exhibit when z of the cats are loins

1 answer:

grigory [225]3 years ago

6 0

Total cats: 8
Total expense: y
Expense per lion: $8,000
Total lions: z
Expense per leopard: $6,000
Total leopards: 8 - z
Therefore, the equation becomes:
total expense = number of lions × expense per lion + number of leopards ×expense per leopard
y = 8000z + 6000(8 - z)
y = 2000z + 48000

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Find the exact value of the expression.<br> tan( sin−1 (2/3)− cos−1(1/7))

Sonja [21]

Answer:

$\tan(a-b)=\frac{2\sqrt{5}-20\sqrt{3}}{5+8\sqrt{15}}$

Step-by-step explanation:

I'm going to use the following identity to help with the difference inside the tangent function there:

$\tan(a-b)=\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}$

Let $a=\sin^{-1}(\frac{2}{3})$ .

With some restriction on $a$ this means:

$\sin(a)=\frac{2}{3}$

We need to find $\tan(a)$ .

$\sin^2(a)+\cos^2(a)=1$ is a Pythagorean Identity I will use to find the cosine value and then I will use that the tangent function is the ratio of sine to cosine.

$(\frac{2}{3})^2+\cos^2(a)=1$

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Subtract 4/9 on both sides:

$\cos^2(a)=\frac{5}{9}$

Take the square root of both sides:

$\cos(a)=\pm \sqrt{\frac{5}{9}}$

$\cos(a)=\pm \frac{\sqrt{5}}{3}$

The cosine value is positive because $a$ is a number between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ because that is the restriction on sine inverse.

So we have $\cos(a)=\frac{\sqrt{5}}{3}$ .

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$\tan(a)=\frac{2\sqrt{5}}{5}$

Let's continue on to letting $b=\cos^{-1}(\frac{1}{7})$ .

Let's go ahead and say what the restrictions on $b$ are.

$b$ is a number in between 0 and $\pi$ .

So anyways $b=\cos^{-1}(\frac{1}{7})$ implies $\cos(b)=\frac{1}{7}$ .

Let's use the Pythagorean Identity again I mentioned from before to find the sine value of $b$ .

$\cos^2(b)+\sin^2(b)=1$

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Subtract 1/49 on both sides:

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Take the square root of both sides:

$\sin(b)=\pm \sqrt{\frac{48}{49}$

$\sin(b)=\pm \frac{\sqrt{48}}{7}$

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This implies:

$\sin(b)=\frac{4\sqrt{3}}{7}$

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Multiplying both top and bottom by 7 gives:

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Let's put everything back into the first mentioned identity.

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$\tan(a-b)=\frac{2\sqrt{5}-20\sqrt{3}}{5+8\sqrt{15}}$

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Step-by-step explanation:

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Answer:

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