Linear Functions Word Problems.pdf - Free Printable

Let's solve both problems step by step.

---

PROBLEM 1: Cricket Problem

We are told that the chirping rate of crickets is a linear function of temperature. We are given two data points:

- At 59°F, chirps = 76 per minute
- At 65°F, chirps = 100 per minute

Let:
- $ T $ = temperature in °F
- $ C $ = chirping rate in chirps per minute

We want to find a linear equation of the form:
$$
C = mT + b
$$
where $ m $ is the slope and $ b $ is the y-intercept (chirp rate intercept).

---

#### A. Write an equation expressing chirping rate in terms of temperature

First, find the slope $ m $:
$$
m = \frac{100 - 76}{65 - 59} = \frac{24}{6} = 4
$$

So, the chirping rate increases by 4 chirps per minute per degree Fahrenheit.

Now use point-slope form with one point, say $ (59, 76) $:
$$
C - 76 = 4(T - 59)
$$
$$
C = 4T - 236 + 76 = 4T - 160
$$

✔ So the equation is:
$$
\boxed{C = 4T - 160}
$$

---

#### B. Predict the chirping rate at 90°F

Plug $ T = 90 $ into the equation:
$$
C = 4(90) - 160 = 360 - 160 = 200
$$

✔ Chirping rate at 90°F is 200 chirps per minute.

---

#### C. Give the temperature if you count 120 chirps per minute

Set $ C = 120 $:
$$
120 = 4T - 160
$$
$$
4T = 280 \Rightarrow T = 70
$$

✔ The temperature is 70°F.

---

#### D. Calculate the temperature intercept. What does this number tell you about the real world of crickets?

The temperature intercept is when $ C = 0 $. Solve for $ T $:
$$
0 = 4T - 160 \Rightarrow T = 40
$$

So, the temperature intercept is 40°F.

This means that according to the model, crickets would stop chirping at 40°F. In reality, this suggests that crickets do not chirp below this temperature — likely because they are too cold. It’s a biological threshold.

✔ Answer: Temperature intercept is 40°F, meaning crickets stop chirping at or below this temperature.

---

#### E. What is the chirping rate intercept? What does this number tell you about the real world of crickets?

The chirping rate intercept is when $ T = 0 $:
$$
C = 4(0) - 160 = -160
$$

So, the chirping rate intercept is -160 chirps per minute.

But this doesn't make sense in the real world — negative chirps are impossible.

✔ This tells us that the model is only valid within a certain temperature range. Extrapolating to $ T = 0 $ gives an unrealistic result. The model is not valid for very low temperatures.

---

#### F. What is a reasonable domain for this problem? Explain.

We know:
- Crickets don’t chirp below ~40°F (from D)
- Realistic cricket activity occurs between roughly 50°F and 90°F
- Above 90°F, crickets might become stressed or stop chirping due to heat

Also, the data starts at 59°F, so we should consider realistic biological limits.

✔ A reasonable domain is:
$$
\boxed{[50^\circ F,\ 90^\circ F]}
$$
or possibly $ [40^\circ F,\ 90^\circ F] $, but since we have no data below 59°F, it's safer to go from 50°F to 90°F.

---

#### G. Sketch the graph of this function

The graph is a straight line with:
- Slope = 4
- Y-intercept = -160 (but not meaningful)
- Passes through points: (59, 76), (65, 100), (90, 200)

Plotting:
- On x-axis: Temperature (°F)
- On y-axis: Chirps per minute
- Draw a line from (50, 40) to (90, 200), but only show where it makes sense.

Note: At $ T = 50 $, $ C = 4(50) - 160 = 200 - 160 = 40 $

So points: (50, 40), (59, 76), (65, 100), (90, 200)

Sketch a straight line through these, labeling axes and showing the domain.

✔ Graph: Linear increasing line from (50, 40) to (90, 200), with labels.

---

## ✔ Summary – Problem 1 Answers:

| Part | Answer |
|------|--------|
| A | $ C = 4T - 160 $ |
| B | 200 chirps/min |
| C | 70°F |
| D | 40°F — temperature at which crickets stop chirping |
| E | -160 chirps/min — not realistic; model fails at low temps |
| F | Domain: $ [50^\circ F, 90^\circ F] $ |
| G | Linear graph from (50,40) to (90,200) |

---

PROBLEM 2: Speed on a Hill Problem

We're told that maximum speed $ S $ (in mph) is a linear function of hill steepness $ s $ (in degrees). We define:
- Going up a hill: positive steepness
- Going down a hill: negative steepness

Given:
- At $ s = 5^\circ $ (uphill), $ S = 55 $ mph
- At $ s = -2^\circ $ (downhill), $ S = 104 $ mph

Let:
- $ S = ms + b $, where $ m $ is the rate of change of speed with respect to steepness

---

#### A. Write an equation expressing maximum speed in terms of steepness

Find slope $ m $:
$$
m = \frac{104 - 55}{-2 - 5} = \frac{49}{-7} = -7
$$

So, for every degree increase in steepness (going up), speed decreases by 7 mph.

Use point $ (5, 55) $:
$$
S - 55 = -7(s - 5)
$$
$$
S = -7s + 35 + 55 = -7s + 90
$$

✔ Equation: $ \boxed{S = -7s + 90} $

---

#### B. How fast could your car go up a 7° hill?

$ s = 7 $:
$$
S = -7(7) + 90 = -49 + 90 = 41
$$

✔ 41 mph

---

#### C. How fast could the car go up a 7° hill?

Wait — this is the same as part B.

But let’s recheck: "How fast could the car go up a 7° hill?" → yes, same as B.

But now check: “How fast could the car go up a 7° hill?” → already answered: 41 mph

Wait — maybe typo? Let's read again.

Actually, part C says:
> C. How fast could the car go up a 7° hill?

Yes, same as B. But perhaps it's a repeat. Or maybe a typo?

Wait — look again:
B: "How fast could your car go up a 7° hill?"
C: "How fast could the car go up a 7° hill?"

Same question. Probably a typo. Maybe C is meant to be "down" a 7° hill?

Let me assume that C is asking for going down a 7° hill, since otherwise it's redundant.

So:
- C. How fast could the car go down a 7° hill?

Then $ s = -7 $:
$$
S = -7(-7) + 90 = 49 + 90 = 139
$$

✔ 139 mph (theoretically, though probably unsafe!)

But wait — is that realistic? Yes, mathematically.

So assuming C is downhill 7°, answer is 139 mph.

But if it's really the same as B, then answer is 41 mph.

But since it's listed twice, likely a typo.

Let’s suppose C is asking for going up a 7° hill → answer is 41 mph.

But let’s keep going.

Wait — actually, looking back:

> C. How fast could the car go up a 7° hill?

It's written twice — probably a mistake in formatting.

But based on context, B and C are the same. Perhaps C was meant to be downhill.

Alternatively, maybe C is asking about uphill, and B is asking about downhill?

No — both say “up”.

Wait — no: B says “up a 7° hill”, C says “up a 7° hill” — same.

Possibility: Typo in problem. But let’s proceed with what's written.

Assume C is also “up a 7° hill” → answer: 41 mph

But that duplicates B.

Alternatively, maybe C is asking for going down a 7° hill.

Let’s assume that C is asking for going down a 7° hill, i.e., $ s = -7 $.

Then:
$$
S = -7(-7) + 90 = 49 + 90 = 139 \text{ mph}
$$

✔ So if C is downhill, answer is 139 mph

But let’s move to D, which is different.

---

#### D. If the maximum speed is 83 miles per hour, what is the steepness of the hill? Are you going up or down? Justify your answer.

Set $ S = 83 $:
$$
83 = -7s + 90
\Rightarrow -7s = -7 \Rightarrow s = 1
$$

So steepness is 1°

Since $ s = 1 > 0 $, you are going up a hill.

✔ Steepness: 1° uphill

---

#### E. What does the speed-intercept equal, and what does it represent?

Speed-intercept is when $ s = 0 $:
$$
S = -7(0) + 90 = 90
$$

So speed-intercept is 90 mph.

This represents the maximum speed on a flat road (0° steepness).

✔ Answer: 90 mph — speed on level ground.

---

#### F. What does the steepness-intercept equal, and what does it represent?

Steepness-intercept is when $ S = 0 $:
$$
0 = -7s + 90 \Rightarrow s = \frac{90}{7} \approx 12.86^\circ
$$

So at ~12.86° uphill, the car can’t go anymore — speed drops to zero.

This represents the maximum incline the car can climb without assistance.

✔ Steepness-intercept: $ \boxed{\frac{90}{7}^\circ \approx 12.86^\circ} $ — steepest hill the car can climb.

---

#### G. What would be a reasonable domain for this function? Explain.

The model is linear, but physically limited.

- Maximum uphill: when $ S = 0 $, $ s = 12.86^\circ $
- Downhill: theoretically, speed increases, but safety limits apply.
- But realistically, cars can’t go down extremely steep hills safely.

From data: