The density, D, of an object is the mass per unit volume, V. That is, D = \dfrac{m}{V}.
A metal rod has fixed mass of 12 \text{ kg} and variable volume and density due to the thermal expansion from the sun.
Find the density of the metal rod with volume 0.2 \text{ m}^3 in \text{ kg/m}^3 .
Find the density of the metal rod with volume 0.625 \text{ m}^3 in \text{ kg/m}^3 .
The population density, D, of a species is the number of organisms, n, per unit area, A \text{ km}^2. That is, D = \dfrac{n}{A}.
A species has a population of 27\,397.
Find the population density across an area of 801 \text{ km}^2 to two decimal places.
Find the population density across an area of 617 \text{ km}^2 to two decimal places.
The pressure, P \text{ kPa}, and volume, V \text{ cm}^3, of a gas at a certain temperature are related by the equation P V = 87.73.
Find the volume, V, when the pressure is 953 \text{ kPa} to two decimal places.
Find the pressure, P, when the volume is 0.08 \text{ cm}^3 to two decimal places.
Hiring a bus costs \$950 regardless of the number of passengers.
Write an equation that relates the number of passengers, n, to the unit price per passenger, p.
How much will each passenger pay if there are 38 passengers?
Xavier is planning on driving 720 \text{ km} .
Write an equation relating Xavier's time, t hours, to his speed, s \text{ km/h} .
How long will it take Xavier to reach his destination if he travels at 90 \text{ km/h}?
How fast would Xavier have to travel to reach his destination in 4 hours?
After a rain storm, a water tank has enough water to cover 300 \text{ m}^2 with 10.2 \text{ mm} of water or 375 \text{ m}^2 with 8.16 \text{ mm} of water.
Write an equation relating the depth of water, D \text{ mm}, to the area of the land, A \text{ m}^2.
Find the depth of water when the area of land is 425 \text{ m}^2.
Valentina wants to save \$30\,000 by saving the same amount, \$P each month.
Write an equation relating the amount, P, that Valentina must save each month to the number of months, N, needed to save.
How long will it take Valentina to save \$30\,000 if she saves \$1500 each month?
How much would Valentina need to save each month if she wants to save \$30\,000 in two months?
The current in Amperes, I, and resistance in Ohms, R, in an electrical circuit have the relationship I R = 15.
Find the current when the resistance is 3 Ohms.
Describe the current as the resistance gets larger and larger.
Describe the current as the resistance gets closer and closer to 0.
A sound wave is measured to have a frequency of 100 \text{ Hz} with a wavelength of 0.3 \text{ m} and a frequency of 50 \text{ Hz} with a wavelength of 0.6 \text{ m}.
Write an equation to model the inverse relationship between the frequency of the wave, f \text{ Hz}, and the wavelength, l metres.
Find the frequency when the wavelength is 0.4 \text{ m} to the nearest two decimal places.
Describe the frequency as the wavelength gets larger and larger.
Describe the frequency as the wavelength gets closer and closer to 0.
Newton's second law states that the force, F, on an object is proportional to its mass, m, and acceleration, a. Newton's third law states that the magnitude of forces on two objects in an interaction is equal. In an interaction between a 12 \text{ kg} object and a 19 \text{ kg} object, the magnitude of the force is measured to be 182.4 \text{ N} .
Write an equation relating the mass to the acceleration of an object in this interaction.
Find the acceleration of the 12 \text{ kg} object.
Find the acceleration of the 19 \text{ kg} object.
A fixed amount of batter can be used to make 220 \text{ cm}^3 of cake.
Write an equation that relates the height of the cake, h \text{ cm} to the cross sectional area, A \text{ cm}^2.
How tall will the cake be in centimetres if the cross-sectional area is 60 \text{ cm}^2? Round your answer to the nearest two decimal places.
How tall will the cake be in centimetres if the cross-sectional area is 3601 \text{ cm}^2? Round your answer to the nearest two decimal places.
Which are more reasonable, the values found in part b, or the values found in part c ?
It takes one software developer 16 hours to complete a task, and it takes ten software developers 1.6 hours to complete the same task.
Write an equation that relates the number of software developers working on the task, n, to the time in hours taken to complete it, t.
How long will the task take if there are 8 software developers?
How long will the task take if there are 64 software developers?
Comment on how reasonable the values in parts b and c are.
A book editor approximates that she will take 50 hours altogether to edit a new book. She wants to minimise the number of days spent on editing the book.
Complete the following table of values:
| \text{Number of hours editing per day } (x) | 2 | 4 | 6 | 8 | 10 |
| \text{Number of days it will take }(y) |
Form an equation for the relationship between the number of hours spent editing each day, x, and the number of days it will take to complete the work, y.
Roxanne can only spend up to 7.5 hours per day on this job. Find the least number of days it would take to finish the job.
Find the largest possible value of x that can be used in this model.
A group of architecture students are given the task of designing the layout of a house with a rectangular floorplan. There are no restrictions on the length and the width of the house, but the floor area must be 120 square metres. Each student will be allocated a rectangle with a different pair of dimensions to any other student.
Complete the table for the various widths given:
| \text{Width in metres } (x) | 5 | 10 | 15 | 20 | 25 |
|---|---|---|---|---|---|
| \text{Length in metres } (y) |
Form an equation for y in terms of x.
As the width of the house increases, what happens to the length of the house?
If the width is 24 \text{ m}, what will be the length of the floor area?
Sketch the graph of the relationship between x and y.
A group of people are trying to decide whether to charter a yacht for a day trip to the Great Barrier Reef. The total cost of chartering a yacht is \$1200. The cost per person if n people embark on the trip is given by C = \dfrac{1200}{n}.
Complete the following table of values:
| n | 1 | 2 | 4 | 6 | 8 | 10 | 12 | 14 |
|---|---|---|---|---|---|---|---|---|
| C \, (\$) |
Sketch the graph of C = \dfrac{1200}{n}.
Alternatively, the day tour costs \$120 per person to run. Using the graph or otherwise, determine how many people will be needed to charter the yacht so that the two options cost the same for each person.