Word Problems and Modeling
When writing linear models from word problems, you need to know what the x and y variables refer to, and what the slope and y-intercept are.
Here are a few helpful hints.
From the given information, identify the numbers asked for.
2. Choose a variable to represent the number(s) described in the problem. (A sketch or chart may be helpful.)
3. Write an equation representing the relationships among the numbers in the problem.
4. Solve the equation and find the required numbers
5. Answer the original question. Check that your answer is reasonable.
The linear model y = mx + b can be used as a model that involves a constant rate of change.
Let's look at simple problems. Write a linear model, in slope-intercept form, to model each situation.
1. You rent a bicycle for $20 plus $2 per hour.
x = hours y = cost
x = $2/hour (constant rate of change)
y = the initial cost is $20.
Model: y = 2x + 20
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2. The temperature is 15º and is expected to fall 2º each hour during the night.
x = hours
y = temp º
x = -2º/hour (constant rate of change)
y = the initial temp is 15º
Model: y = -2x + 15
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3. The cost for 7 dance lessons is $82. The cost for 11 lessons is $122.
Find the total cost, C, for lessons, L.
Notice we do not have a constant rate of change, so we must find it.
We are given two ordered pairs (7, 82) and (11, 122).
Find m: (122 - 82) / (11 - 7) = 40/4 = 10 (rate of change)
C = 10L + ? Plug in a given point -> (7, 82)
82 = 10 (7) + ?
82 - 70 = ?
12 = ?
Model: C = 10L + 12
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4. It is 76º F at the 6000-foot level of a mountain, and 49º F at the 12,000-foot level of the mountain.
Find the temperature, T, at an elevation, e, on the mountain, where e is thousands of feet.
Is there a constant rate of change? No, but we do have two ordered pairs: (6, 76) and (12, 49).
Find the rate of change, m. (49 - 76)/(12 - 6) = -27/6 = -9/2
T = -9/2e + ? Plug in a given point (6, 76)
76 = -9/2 (6) + ?
76 = -27 + ?
103 = ?
Model: T = -9/2e + 103