- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
The definition of the derivative practice question 7
A horse’s daily ration is kilograms of oats and kilograms of hay. Oats supply 12 megajoules (MJ) of energy per kilogram and hay supplies 8 MJ per kilogram, and the ration must supply exactly 72 MJ, so . Plotted with on the horizontal axis and on the vertical axis, these rations lie on a line. Find its slope.
Give the answer in kilograms of hay per kilogram of oats.
With the energy fixed, does more oats mean more hay or less? Which form of the line’s equation shows its slope directly?
Solve the equation for to put it in the form . The slope is , the coefficient of .
A fraction, or a decimal to two decimal places.
Theory UBC Math 100 §3.1Key results
Worked solution
Strategy
Put the line in slope-intercept form by solving for y; the coefficient of x is the slope. Read as a rate, it says how much the hay must change for each extra kilogram of oats while the energy stays at 72 MJ.
Working
Move the oats term across, then divide by 8.
Check with two rations on the line: hay alone, (0, 9), and oats alone, (6, 0). Rise over run agrees.
Answer
The answer is negative 3/2 kilograms of hay per kilogram of oats: each extra kilogram of oats replaces one and a half kilograms of hay.
Check
The rations (2, 6) and (4, 3) both give 72 MJ, and between them hay falls 3 kg as oats rise 2 kg.
Where this goes wrong
Dropping the minus sign and answering 3/2. A kilogram of oats carries as much energy as 1.5 kg of hay, so each extra kilogram of oats means 1.5 kg less hay: the line falls.
Leans on
Slope-intercept form of a line, from Functions, graphs and inverses
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