Question

A hose is left running for 240 minutes to 2 significant figures.

The amount of water coming out of the hose each minute is 2.1 metres to 2 significant figures.

Calculate the power and upper bounds for the total amount of water that comes out of the hose.

Answer 1

The total amount of water that comes out of the hose is approximately between 492 litres and 515 litres.

To calculate the lower and upper bounds of the total amount of water that comes out of the hose, we can use the concept of significant figures and propagation of errors.

1. Calculating Total Amount of Water:

The total amount of water (V) can be found by multiplying the time (t) by the rate of water flow per minute (r).

`\[ V = t * r \]`

Given:

`\( t = 240 \ minutes \) (2 significant figures)`

`\( r = 2.1 \ litres/minute \) (2 significant figures)`

`\[ V = 240 \ minutes * 2.1 \ litres/minute \]`

`\[ V = 504 \ litres \]`

2. Determining Lower and Upper Bounds:

To find the lower and upper bounds, we consider the possible errors in the measurements. Since both time and rate have two significant figures, we can use the absolute uncertainties for each value.

For time:
`\[ \Delta t = \pm0.5 \ minutes \]`

For rate:
`\[ \Delta r = \pm0.05 \ litres/minute \]`

Using the formula for the product of two variables, the absolute uncertainty in the total volume (\( \Delta V \)) is given by:

`\[ \Delta V = √((t * \Delta r)^2 + (r * \Delta t)^2) \]`

Substituting the values:

`\[ \Delta V = √((240 \ minutes * 0.05 \ litres/minute)^2 + (2.1 \ litres/minute * 0.5 \ minutes)^2) \]`

`\[ \Delta V \approx \pm 12.004 \ litres \]`

3. Final Result:

The lower and upper bounds of the total amount of water are:

`\[ 504 \ litres - 12.004 \ litres = 491.996 \ litres \]`

`\[ 504 \ litres + 12.004 \ litres = 515.004 \ litres \]`