# Upper and lower boundary and difficult examples

Example 1

The formula for the height a ball can reach when thrown was given as:

h=(v^2)/(2g)

Where:

v= velocity

g=gravity

h=height

If the velocity is 13.6\ m//s correct to 1 decimal place and gravity is 9.8\ m//s^2 correct to 1 decimal place. What is the lower boundary also correct to 1 D.P.?

NOTE:

The lower boundary of a division would be:

Lower\ boundary=(Small)/(Big)=(Lower\ boundary\ A)/(Upper\ boundary\ B)

First find the lower boundary of v=13.6\ m//s correct to 1 D.P.

13.6\ m//s to one decimal place

13.ul6 underline the digit (the 1^(st) decimal)

13.ul6\0 look next door

color(red)0     5 or more raise the score so 13.55 would raise to 13.6

Now find the upper boundary of g=9.8\ m//s^2 correct to 1 D.P.

9.8\ m//s^2 to one decimal place

9.ul8 underline the digit (the 1^(st) decimal)

9.ul8\0 look next door

color(red)0     5 or more raise the score so 9.8499 would ignore and stay at 9.8

This is simplified to 9.85

So the lower boundary of

h=(v^2)/(2g)

is

h=13.55^2/(2times9.85)

h=9.319923

But rounded to 1 D.P. h=9.3\ metres

Example 2

You are borrowing a friends car and you need to know how expensive the fuel consumption is i.e. miles/gallon (mpg) of the car. Your friend left a note saying that the car travelled 1500miles and she used 31 gallons of fuel.
The mileage was correct to the nearest 100 miles and the gallons used was correct to the nearest gallon.
What are the upper and lower boundaries of the fuel consumption miles/gallon (mpg)?

First find the formula used i.e.

1500\ mi\l\es=31\ gallons

x\ mi\l\es=1\ gallon

(see our notes on % percentages to understand how this works)

Put a division line in

1500/x=31/1

Put x as the subject

x=(1500times1)/31

x=1500/31=48.39 correct to 2 D.P.

48.39 miles/gallon would be the answer if there were no boundaries.

But because there are boundaries it is:

(1500\ t\o\ n\e\arest\ 100\ mi\l\es)/(31\ t\o\ n\e\arest\ gallon)

Find the upper and lower boundary of the 1500 mile to nearest 100 miles.

1500 to nearest 100 miles

1ul5\00 underline the digit (the 100's unit)

1ul5\color(red)0\0 look next door

0        5 or more raise the score so 1450 would raise to 1500

0        4 or less just ignore 1549... it would ignore and stay at 1500

This is simplified to 1550

So the upper and lower boundary of 1500 is 1550 and 1450.

Now find the upper and lower boundaries of 31 gallons to the nearest gallon.

31      to nearest 1 gallon

3ul1      underline the digit (the 1's unit)

3ul1.\color(red)0 look next door

0        5 or more raise the score so 30.5 would raise to 31

0        4 or less just ignore 31.499... it would ignore and stay at 31

This is simplified to 31.5

So the upper and lower boundary of 31 is 31.5 and 30.5

Summary

The upper and lower boundary of 1500 is 1550 and 1450

The upper and lower boundary of 31 is 31.5 and 30.5

NOTE:

The upper boundary of division is (Big)/(Small)=(Upper\ boundary\ A)/(Lower\ boundary\ B)

So the upper boundary is 1550/30.5=50.82\ mi\l\es//gallon correct to 2 D.P.

The lower boundary of division is (Small)/(Big)=(Lower\ boundary\ A)/(Upper\ boundary\ B)

So the lower boundary is 1450/31.5=46.03\ mi\l\es//gallon correct to 2 D.P.

Answer: The Upper and lower boundaries are 50.82 and 46.03\ mi\l\es//gallon