Leaving Cert Mathematics Number Systems: (a) The mean distance from the e

(a) The mean distance from the earth to the sun is 149 597 871 km - Leaving Cert Mathematics - Question Question (a) and (b) - 2013

Question Question (a) and (b)

(a) The mean distance from the earth to the sun is 149 597 871 km. Write this number in the form $a \times 10^n$, where $1 < a < 10$ and $n \in \mathbb{Z}$, correct ... show full transcript

Worked Solution & Example Answer:(a) The mean distance from the earth to the sun is 149 597 871 km - Leaving Cert Mathematics - Question Question (a) and (b) - 2013

Step 1

Write the mean distance from the Earth to the Sun in scientific notation

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Answer

To convert 149,597,871 km into scientific notation, we first express it as follows:

$149597871 = 1.5 \times 10^8$

Here, $1.5$ is between 1 and 10 and $8$ is an integer.

Step 2

Write each of the numbers below as a decimal correct to two decimal places

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Answer

Variable	Calculation	Result
A	$\sqrt{5}$	2.24
B	$\frac{243}{85}$	2.86
C	$\tan(70°)$	2.75
D	$\frac{3\pi}{4}$	2.36
E	$\frac{250}{100}$	2.50
F	$\left(1 + \frac{1}{10}\right)^{10}$	2.59

Step 3

Mark 5 of the numbers in the table on the number line below and label each number clearly

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Answer

On the number line provided, we can mark the following values:

2 at position 2
2.24 at position A
2.50 at position F
2.75 at position E
2.86 at position B

The completed number line would look like this:

<----|----|---A---|---B---|-----|---F---|---2----|---2.5---3--->
     2   2.24  2.86   2.5   2.75

Step 4

Solve the equation $27^{n} = 3^{n + 10}$

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Answer

To solve for $n$ , we start by rewriting $27$ as $3^3$ :

$27^{n} = (3^{3})^{n} = 3^{3n}$

Now we can equate the exponents: $3^{3n} = 3^{n + 10}$

This gives us: $3n = n + 10$

Subtracting $n$ from both sides: $2n = 10$

Dividing by 2: $n = 5$

(a) The mean distance from the earth to the sun is 149 597 871 km - Leaving Cert Mathematics - Question Question (a) and (b) - 2013

Worked Solution & Example Answer:(a) The mean distance from the earth to the sun is 149 597 871 km - Leaving Cert Mathematics - Question Question (a) and (b) - 2013

Write the mean distance from the Earth to the Sun in scientific notation

Write each of the numbers below as a decimal correct to two decimal places

Mark 5 of the numbers in the table on the number line below and label each number clearly

Solve the equation $27^{n} = 3^{n + 10}$

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