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3 years ago

Find the number of positive integers n is less than or equal to 1000 such that 15n is a perfect square.

Mathematics

1 answer:

boyakko [2]3 years ago

3 0

Answer:

8 positive integers.

Step-by-step explanation:

One value of n would be 15 because 225 = 15^2.

Other values are 225 * n where 15n <= 1000 and n is a perfect square.

So n = 4 gives us 225* 4 = 900 which is a perfect square and 15*4 = 60.

n = 9 gives us 225 * 9 which is a perfect square and 15*9 = 135.

n = 16 gives us 225*16 and 15*16 = 240 , so OK.

n = 25 gives a perfect square and and 15*25 = 375 - so OK.

n = 36 gives a perfect square and 15*36 = 540 - so OK.

n = 49 gives a perfect square and 15*49 = 735 - so OK.

n = 64 gives a perfect square and 15*64 = 960 - so ok.

n = 81 gives a perfect square and 15 * 81 = 1215 so NOT ok.

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This means that the randomization will have the null hypothesis saying that the proportion of students who will choose maths as their favourite subject = 0.1

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2 years ago

Suppose you have a light bulb that emits 50 watts of visible light. (Note: This is not the case for a standard 50-watt light bul

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Step-by-step explanation:

The apparent brightness of a star is

$\bf B=\displaystyle\frac{L}{4\pi d^2}$

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Hence the apparent brightness of Alpha Centauri A is

$\bf B=\displaystyle\frac{1.519}{4\pi (4.37)^2}=0.006329728$

According to the inverse square law for light intensity

$\bf \displaystyle\frac{I_1}{I_2}=\displaystyle\frac{d_2^2}{d_1^2}$

where

$\bf I_1=$ light intensity at distance $\bf d_1$

$\bf I_2=$ light intensity at distance $\bf d_2$

Let $\bf d_2$ be the distance we would have to place the 50-watt bulb, then replacing in the formula

$\bf \displaystyle\frac{0.006329728}{50}=\displaystyle\frac{d_2^2}{(4.37)^2}\Rightarrow\\\\\Rightarrow d_2^2=\displaystyle\frac{0.006329728*(4.37)^2}{50}\Rightarrow d_2^2=0.002417564\Rightarrow\\\\\Rightarrow d_2=\sqrt{0.002417564}=\boxed{0.049168726\;ly}$

Remark: It is worth noticing that Alpha Centauri A, though is the nearest star to the Sun, is not visible to the naked eye.

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