Question Statement

A vacuum pump removes $\frac{1}{5}$ of the air from a sealed container on each stroke of its piston. What percentage of the original air remains after 5 strokes?

Background and Explanation

This problem can be solved using the principles of a geometric sequence. After each stroke, a constant fraction of the remaining air is left in the container. This constant fraction acts as the common ratio ( $r$ ) of the sequence, where each term represents the amount of air left after each stroke.

Solution

To find the percentage of air remaining after 5 strokes, we first need to determine the amount of air left after each stroke as a fraction of the original amount.

Air Remaining After the 1st Stroke ( $a_{1}$ ): If the pump removes $\frac{1}{5}$ of the air, the fraction of air that remains is: $1 - \frac{1}{5} = \frac{4}{5}$ So, after the first stroke, $\frac{4}{5}$ of the original air is left. This is the first term of our sequence, $a_{1} = \frac{4}{5}$ .
Air Remaining After the 2nd Stroke ( $a_{2}$ ): The pump removes $\frac{1}{5}$ of the remaining air. This means $\frac{4}{5}$ of the air from the previous stroke is left. Amount left = (Amount after 1st stroke) $\times \frac{4}{5} = \frac{4}{5} \times \frac{4}{5} = (\frac{4}{5})^{2}$
Identify the Geometric Sequence: The fraction of air remaining after each stroke forms a geometric sequence:
- $a_{1} = \frac{4}{5}$
- $a_{2} = (\frac{4}{5})^{2}$
- $a_{3} = (\frac{4}{5})^{3}$ The first term is $a_{1} = \frac{4}{5}$ and the common ratio is $r = \frac{4}{5}$ .
Find the Air Remaining After the 5th Stroke ( $a_{5}$ ): We need to find the 5th term of this sequence ( $n = 5$ ). We use the formula for the $n^{t h}$ term of a geometric sequence: $a_{n} = a_{1} \cdot r^{n - 1}$ .

$a_{5} = (\frac{4}{5}) \cdot (\frac{4}{5})^{5 - 1}$ $a_{5} = (\frac{4}{5}) \cdot (\frac{4}{5})^{4}$ $a_{5} = (\frac{4}{5})^{5}$ $a_{5} = \frac{4 ^{5}}{5 ^{5}} = \frac{1024}{3125}$
Convert the Fraction to a Percentage: To express this as a percentage, we convert the fraction to a decimal and multiply by 100.

$\frac{1024}{3125} \approx 0.32768$

$0.32768 \times 100% = 32.768%$

Rounding to one decimal place, approximately 32.8% of the air remains after 5 strokes.

Key Formulas or Methods Used

The core formula used is for the $n^{t h}$ term of a geometric sequence:

$a_{n} = a_{1} \cdot r^{n - 1}$

Where:

$a_{n}$ is the amount remaining after the $n^{t h}$ stroke.
$a_{1}$ is the amount remaining after the first stroke.
$r$ is the common ratio (the fraction of air that remains after each stroke).
$n$ is the number of strokes.

Summary of Steps

Calculate the fraction of air remaining after one stroke: $1 - \frac{1}{5} = \frac{4}{5}$ .
Identify this as a geometric sequence where the first term $a_{1} = \frac{4}{5}$ and the common ratio $r = \frac{4}{5}$ .
Set the term number to $n = 5$ to find the amount remaining after 5 strokes.
Apply the $n^{t h}$ term formula: $a_{5} = a_{1} \cdot r^{4} = (\frac{4}{5}) \cdot (\frac{4}{5})^{4} = (\frac{4}{5})^{5}$ .
Calculate the value and convert the fraction $\frac{1024}{3125}$ to a percentage, which is approximately 32.8%.

Solution

To find the percentage of air remaining after 5 strokes, we first need to determine the amount of air left after each stroke as a fraction of the original amount.

Air Remaining After the 1st Stroke ( $a_{1}$ ): If the pump removes $\frac{1}{5}$ of the air, the fraction of air that remains is: $1 - \frac{1}{5} = \frac{4}{5}$ So, after the first stroke, $\frac{4}{5}$ of the original air is left. This is the first term of our sequence, $a_{1} = \frac{4}{5}$ .

Air Remaining After the 2nd Stroke ( $a_{2}$ ): The pump removes $\frac{1}{5}$ of the remaining air. This means $\frac{4}{5}$ of the air from the previous stroke is left. Amount left = (Amount after 1st stroke) $\times \frac{4}{5} = \frac{4}{5} \times \frac{4}{5} = (\frac{4}{5})^{2}$

Identify the Geometric Sequence: The fraction of air remaining after each stroke forms a geometric sequence:

$a_{1} = \frac{4}{5}$
$a_{2} = (\frac{4}{5})^{2}$
$a_{3} = (\frac{4}{5})^{3}$ The first term is $a_{1} = \frac{4}{5}$ and the common ratio is $r = \frac{4}{5}$ .

Find the Air Remaining After the 5th Stroke ( $a_{5}$ ): We need to find the 5th term of this sequence ( $n = 5$ ). We use the formula for the $n^{t h}$ term of a geometric sequence: $a_{n} = a_{1} \cdot r^{n - 1}$ .

$a_{5} = (\frac{4}{5}) \cdot (\frac{4}{5})^{5 - 1}$ $a_{5} = (\frac{4}{5}) \cdot (\frac{4}{5})^{4}$ $a_{5} = (\frac{4}{5})^{5}$ $a_{5} = \frac{4 ^{5}}{5 ^{5}} = \frac{1024}{3125}$

Convert the Fraction to a Percentage: To express this as a percentage, we convert the fraction to a decimal and multiply by 100.

$\frac{1024}{3125} \approx 0.32768$

$0.32768 \times 100% = 32.768%$

Rounding to one decimal place, approximately 32.8% of the air remains after 5 strokes.

Key Formulas or Methods Used

The core formula used is for the $n^{t h}$ term of a geometric sequence:

a_{n} = a_{1} \cdot r^{n - 1}

Where:

a_{n}

is the amount remaining after the

n^{t h}

stroke.

a_{1}

is the amount remaining after the first stroke.

r

is the common ratio (the fraction of air that remains after each stroke).

n

is the number of strokes.

Summary of Steps

Calculate the fraction of air remaining after one stroke:

1 - \frac{1}{5} = \frac{4}{5}

Identify this as a geometric sequence where the first term

a_{1} = \frac{4}{5}

and the common ratio

r = \frac{4}{5}

Set the term number to

n = 5

to find the amount remaining after 5 strokes.

Apply the $n^{t h}$ term formula:

a_{5} = a_{1} \cdot r^{4} = (\frac{4}{5}) \cdot (\frac{4}{5})^{4} = (\frac{4}{5})^{5}

Calculate the value and convert the fraction

\frac{1024}{3125}

to a percentage, which is approximately 32.8%.

4.8 Q-5

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

4.8 Q-5

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps