Write each of the following values in the form $\alpha \times 10^n$ where $1 \leq \alpha < 10$ and $n \in \mathbb{Z}$. (i) 1200 (ii) 0.27 A falcon can dive at a speed of up to 120 miles per hour. 1 mile is approximately 1.6 kilometres. Use this to work out how long it would take the falcon to travel 100 metres, when diving at this speed. Give your answer in seconds, correct to one decimal place. The diagram below shows the graphs of the functions $k(x)$ and $m(x)$, for $0 \leq x \leq 5$, $x \in \mathbb{R}$. Use the graphs to estimate each of the following, for $0 \leq x \leq 5$: (i) the two values of $x$ for which $m(x) = 0$ (ii) the range of values of $x$ for which $k(x)$ is less than $m(x)$.

Leaving Cert Mathematics Number Systems: Write each of the following valu

Write each of the following values in the form $\alpha \times 10^n$ where $1 \leq \alpha < 10$ and $n \in \mathbb{Z}$ - Leaving Cert Mathematics - Question 5 - 2022

Question 5

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Write each of the following values in the form $\alpha \times 10^n$ where $1 \leq \alpha < 10$ and $n \in \mathbb{Z}$. (i) 1200 (ii) 0.27 A falcon can dive at a s... show full transcript

Worked Solution & Example Answer:Write each of the following values in the form $\alpha \times 10^n$ where $1 \leq \alpha < 10$ and $n \in \mathbb{Z}$ - Leaving Cert Mathematics - Question 5 - 2022

Step 1

Write in the form $\alpha \times 10^n$ for 1200

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Answer

To express 1200 in the form $\alpha \times 10^n$ , we can write it as:

$1200 = 1.2 \times 10^3$

This transformation makes sure that $1 \leq \alpha < 10$ .

Step 2

Write in the form $\alpha \times 10^n$ for 0.27

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Answer

For the value 0.27, we can write it as:

$0.27 = 2.7 \times 10^{-1}$

This is within the required range of $\alpha$ .

Step 3

Use this to work out how long it would take the falcon to travel 100 metres.

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Answer

First, convert the falcon's speed from miles per hour to metres per second:

Speed of the falcon: 120 miles per hour.
- Convert to kilometres per hour: 120 miles x 1.6 km/mile = 192 km/h.
- Convert to metres per second: (192 \text{ km/h} = \frac{192000 ext{ m}}{3600 ext{ s}} = 53.33 \text{ m/s}).
Calculate the time to travel 100 metres:
- Using the formula ( \text{Time} = \frac{\text{Distance}}{\text{Speed}} ): $\text{Time for 100 m} = \frac{100 ext{ m}}{53.33 \text{ m/s}} \approx 1.875 ext{ seconds}$
- So, rounded to one decimal place, the time is approximately 1.9 seconds.

Step 4

the two values of x for which m(x) = 0

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Answer

From the graph, we can observe the intersections of the curve $m(x)$ with the x-axis. The estimated values for ( x ) where ( m(x) = 0 ) are:

$x = 1\text{ or } x = 4$ .

Step 5

the range of values of x for which k(x) is less than m(x)

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Answer

By analyzing the graph, we see that $k(x)$ is less than $m(x)$ between the intervals from:

$2 < x < 3.5$

Therefore, for these values, ( k(x) < m(x) ).

Write each of the following values in the form $\alpha \times 10^n$ where $1 \leq \alpha < 10$ and $n \in \mathbb{Z}$ - Leaving Cert Mathematics - Question 5 - 2022

Worked Solution & Example Answer:Write each of the following values in the form $\alpha \times 10^n$ where $1 \leq \alpha < 10$ and $n \in \mathbb{Z}$ - Leaving Cert Mathematics - Question 5 - 2022

Write in the form $\alpha \times 10^n$ for 1200

Write in the form $\alpha \times 10^n$ for 0.27

Use this to work out how long it would take the falcon to travel 100 metres.

the two values of x for which m(x) = 0

the range of values of x for which k(x) is less than m(x)

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