4.09 Gradients in context
The following graph shows the distance Natalia swam during a recent open water swim:
What is the gradient of the line?
What does the gradient of the line represent in terms of the situation?
The following graph shows the number of batches of cookies produced in a factory:
What is the gradient of the line?
What does the gradient of the line represent in terms of the situation?
Deborah left for a road trip at midday. The following graph shows the total distance travelled (in kilometres), t hours after midday:
How far has the car travelled after 8 hours?
Find the gradient.
Describe what the gradient of the line represents in context.
In 2018, Japan's birthrate was 9 people for every 1000 in the population, while Canada's birthrate was 11 people for every 1000 in the population.
Compute for the number of births expected in that year for the following towns, correct to one decimal place if necessary:
There are 3 apartments per floor of a building. How many apartments are there over 8 floors?
Valerie counts 12 heart beats over a 10 second interval. What is her heart rate in beats per minute (\text{bpm})?
David and Xavier are saving money for a vacation. David knows that he can represent his savings over time using the equation y = \dfrac{13}{2} x, where x represents the number of days and y represents the savings.
Xavier saves \$10 every 3 days. Complete the table showing Xavier's savings over time:
| Days | 6 | 9 | 12 | |
|---|---|---|---|---|
| Dollars saved | 10 | 20 | 30 |
Comparing the graphs of each person's savings, how can you tell who is saving more per day?
Using the graph, who has the greater rate of savings?
Alicia has concluded that the rate of change is greater in f\left(x\right) than g\left(x\right) because it is higher on the given graph.
Explain and correct her error.
Consider the following ramp:
Find the gradient of this skateboard ramp if it rises 0.9 \text{ m} above the ground and runs 1 \text{ m} horizontally at the base.
The ramp can only be used as a 'beginner’s ramp' if for every 1 \text{ m} horizontal run, it has a rise of at most 0.5 \text{ m}. Can it be used as a 'beginner’s ramp'?
A certain ski resort has two ski runs as shown in the diagram:
Find the gradient of Run A. Round your answer to two decimal places.
Find the gradient of Run B. Round your answer to two decimal places.
Which run is steeper?
A carpenter charges a callout fee of \$150 plus \$45 per hour.
Write a linear equation to represent the total amount charged, y, by the carpenter as a function of the number of hours worked, x.
State the gradient of the function.
Describe what the gradient of the line represents in context.
Mohamad is taking his new Subaru out for a drive. He had only driven 50 miles in it before and is now driving it down the highway at 75\text{ mi/h} .
Write an equation to represent the total distance, y, that Mohamad had driven in his Subaru as a function of the number of hours, x.
State the gradient of the function.
Describe what the gradient of the line represents in context.
Mario is running a 100 \text{ km} ultramarathon at an average speed of 9 \text{ km/h}.
Write an equation to represent the distance Mario has left to run, y, as a function of the number of hours since the start, x.
State the gradient of the function.
Describe what the gradient of the line represents in context.
A particular restaurant has a fixed weekly cost of \$1300 and receives an average of \$16 from each customer.
Write an equation to represent the net profit, y, of the restaurant for the week as a function of the number of customers, x.
Find the gradient of the function.
Describe what the gradient of the line represents in context.
A dam used to supply water to the neighboring town had the following data recorded for its volume over a number of months.
| Month | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Volume (billions of litres) | 134 | 117 | 100 | 83 |
At what rate is the volume of water in the dam changing?
Petrol costs a certain amount per litre. The table shows the cost of various amounts of petrol in dollars:
| \text{Number of litres }(x) | 0 | 10 | 20 | 30 | 40 |
|---|---|---|---|---|---|
| \text{Cost of petrol }(y) | 0 | 16.40 | 32.80 | 49.20 | 65.60 |
Find the cost of petrol per litre.
Write an equation linking the number of litres of petrol pumped, x, and the cost of the petrol, y.
Explain the meaning of the gradient in this context.
Calculate the cost of 47 \text{ L} petrol.
A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The following table shows the depth of the diver over 5 minutes.
| \text{Number of minutes passed }(x) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Depth of diver in meters }(y) | 0 | 1.4 | 2.8 | 4.2 | 5.6 |
What is the increase in depth each minute?
Write an equation for the relationship between the number of minutes passed, x, and the depth, y, of the diver.
Explain the meaning of the gradient in this context.
At what depth would the diver be after 6 minutes?
How long does the diver take to reach 12.6 \text{ m} beneath the surface?
A paratrooper falls to the ground along a diagonal line. His fall begins 1157 \text{ m} above the ground, and the line he follows has a gradient of 1.3. That is, he falls 1.3 \text{ m} vertically for every 1 \text{ m} he moves across horizontally.
How far horizontally across the ground does he land from his initial position in the sky?
The graph shows the conversion between temperatures in Celsius and Fahrenheit:
Use the graph to convert 10 \degree \text{C} into Fahrenheit.
0 \degree \text{C} is 32 \degree \text{F}. Hence, for every 1 \degree \text{C} increase, by how much does the Fahrenheit temperature increase?
Would 80 \degree \text{F} be above or below normal body temperature (approximately 37 \degree \text{C})?
Write the rule for conversion between Celsius \left(\text{C}\right) and Fahrenheit \left(\text{F}\right).
Convert 35 \degree \text{C} into Fahrenheit.