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3 years ago

28. A number has 4 and 5 as factors. What other numbers must be factors? Why? What is the smallest number

Mathematics

1 answer:

sweet [91]3 years ago

5 0

Answer:

2, 10.

Smallest number = 20.

Step-by-step explanation:

It must have 2 as a factor because 4 is a factor, and 2 is a factor of 4.

Also 4 *5 = 20 so 10 must be a factor.

The smallest number possible is 20.

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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.

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

$\frac{4}{9}+\cos^2(a)=1$

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

This means that $\tan(a)=\frac{\frac{2}{3}}{\frac{\sqrt{5}}{3}}$ .

Multiplying numerator and denominator by 3 gives us:

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

Rationalizing the denominator by multiplying top and bottom by square root of 5 gives us:

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

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 $b$ .

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

$(\frac{1}{7})^2+\sin^2(b)=1$

$\frac{1}{49}+\sin^2(b)=1$

Subtract 1/49 on both sides:

$\sin^2(b)=\frac{48}{49}$

Take the square root of both sides:

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

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

$\sin(b)=\pm \frac{\sqrt{16}\sqrt{3}}{7}$

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

So since $b$ is a number between $0$ and $\pi$ , then sine of this value is positive.

This implies:

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

So $\tan(b)=\frac{\sin(b)}{\cos(b)}=\frac{\frac{4\sqrt{3}}{7}}{\frac{1}{7}}$ .

Multiplying both top and bottom by 7 gives:

$\frac{4\sqrt{3}}{1}= 4\sqrt{3}$ .

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

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

$\tan(a-b)=\frac{\frac{2\sqrt{5}}{5}-4\sqrt{3}}{1+\frac{2\sqrt{5}}{5}\cdot 4\sqrt{3}}$

Let's clear the mini-fractions by multiply top and bottom by the least common multiple of the denominators of these mini-fractions. That is, we are multiplying top and bottom by 5:

$\tan(a-b)=\frac{2 \sqrt{5}-20\sqrt{3}}{5+2\sqrt{5}\cdot 4\sqrt{3}}$

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

4 0

3 years ago

Zolt Inc. (zolt) stock is selling for $8.22 per share with an EPS of .45 cents per share. What is Zolt's PE ratio? (Round to nea

Umnica [9.8K]

Answer:

18.3 ( when EPS is 45 cents), otherwise there is none of these

Step-by-step explanation:

PE = Current Price / EPS

current Price = $ 8.22 = 822 cents

EPS = .45 cents (?) ... is it 45 cents ?

if it is 45 cents : PE = 822 / 45 = 18.26 ≈ 18.3

7 0

3 years ago

102 is 126% of what number<br><br> 128.52<br> ≈ 8.10 <br> ≈ 80.95 <br> 12.85

disa [49]

Step-by-step explanation:

102*100=10200

10200÷126=80.95

So the third one is correct

4 0

2 years ago

Draw the multiplication table on the P=(3,5,7,9) in module 12

Lisa [10]

Answer:

Find the attached file for the solution

Step-by-step explanation: To draw the multiplication table on the P=(3,5,7,9) in module 12, create the table where all the given parameters will be at the top of horizontal axis and vertical axis,

When multiply by each other, any value that is below 12 will be written down while the value greater than 12 will be divided by 12 and the remainder will be written down.

Find the attached file for the solution and table.

8 0

2 years ago

Jorge bought 6 large balloons for a party. This was 25% of all the balloons he bought. How many total balloons did Jorge buy sho

riadik2000 [5.3K]

Answer:

The answer would be 24 balloons in total.

Step-by-step explanation:

Jorge bought 6 large balloons, which was 25% of ALL the balloons he bought.

Meaning that since 6 is 25% of the number 24, he bought 24 balloons in total.

PROOF:

25% * 24 = 6

24 / 6 = 4

6 * 4 = 24

<h2>Mark me Brainliest Please.</h2>

8 0

2 years ago