transformations - Insert Clever Math Pun Here - Free Printable

Let’s solve each problem one by one. We’ll find the linear equation for each situation — that means we’re looking for a starting value (like a base cost or original length) and then how much it changes per unit (like per km, per gram, per minute, etc.).

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Problem 1: Taxi fare in NYC

We know:
- 1 km → $1.70
- 3 km → $3.30

Let’s think of this as:
Total fare = basic cost + (cost per km × number of km)

So let’s call:
- Basic cost = b
- Cost per km = m

Then:
Equation 1: b + 1m = 1.70
Equation 2: b + 3m = 3.30

Subtract Equation 1 from Equation 2:
(b + 3m) - (b + 1m) = 3.30 - 1.70
→ 2m = 1.60
→ m = 0.80

Now plug back into Equation 1:
b + 0.80 = 1.70
→ b = 0.90

So for a 2 km trip:
Fare = 0.90 + 0.80 × 2 = 0.90 + 1.60 = $2.50

Fill in blanks:
The basic cost for a taxi ride is $0.90 and each kilometer costs $0.80 extra.

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Problem 2: Spring stretch with mass

Given:
- 40g → 6.2 cm
- 60g → 6.7 cm

Let:
- Original length = L
- Stretch per gram = s

Equations:
L + 40s = 6.2
L + 60s = 6.7

Subtract first from second:
(L + 60s) - (L + 40s) = 6.7 - 6.2
→ 20s = 0.5
→ s = 0.025 cm per gram

Plug back:
L + 40×0.025 = 6.2
→ L + 1.0 = 6.2
→ L = 5.2 cm

Now for 90g:
Length = 5.2 + 90×0.025 = 5.2 + 2.25 = 7.45 cm

Fill in blanks:
The original length of the spring is 5.2 cm and each added gram adds 0.025 cm to the length.

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Problem 3: Metal rod expansion with temperature

At 15°C → 72.6 cm
At 95°C → 72.8 cm

Change in temp = 95 - 15 = 80°C
Change in length = 72.8 - 72.6 = 0.2 cm

So per degree: 0.2 / 80 = 0.0025 cm per °C

Starting length at 0°C? Wait — actually, the question says “each degree in temperature adds ___ to the starting length of ___”

But note: they give us length at 15°C, not 0°C. So maybe “starting length” here means at 15°C? Let’s check.

Actually, re-read: “Each degree in temperature adds ___ to the starting length of ___.”

It probably means: from any starting point, each degree adds X cm. And the “starting length” refers to the length at some reference — but since they don’t specify 0°C, perhaps they mean the length at 15°C is the “starting” for calculation? But that doesn’t make sense because then at 75°C, which is 60° above 15°C, we’d add 60 × 0.0025 = 0.15 cm → 72.6 + 0.15 = 72.75 cm.

Wait — let’s do it properly.

Let’s define:
Length = L₀ + k × T, where T is temperature in °C.

We have two points:
T=15 → L=72.6
T=95 → L=72.8

So slope k = (72.8 - 72.6)/(95 - 15) = 0.2/80 = 0.0025 cm/°C

Now, what is L₀? Plug in T=15:

72.6 = L₀ + 0.0025×15
72.6 = L₀ + 0.0375
L₀ = 72.6 - 0.0375 = 72.5625 cm

But the question says: “Each degree in temperature adds ___ to the starting length of ___.”

This phrasing is ambiguous. Maybe they mean: from the given starting point (15°C), each degree adds 0.0025 cm, and the “starting length” is 72.6 cm? That would be simpler for a student.

Let’s assume that’s what they want — because otherwise we’d need to calculate from 0°C, which isn’t mentioned.

So if we take 15°C as “starting”, then:

At 75°C, that’s 75 - 15 = 60 degrees higher.

Add: 60 × 0.0025 = 0.15 cm

New length = 72.6 + 0.15 = 72.75 cm

Fill in blanks:
Each degree in temperature adds 0.0025 cm to the starting length of 72.6 cm.

*(Note: If they meant absolute starting at 0°C, answer would be different, but context suggests using 15°C as reference since it’s given first.)*

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Problem 4: Phone call toll

2 minutes → $0.79
5 minutes → $1.60

Let:
Basic cost = b
Cost per minute = m

Equations:
b + 2m = 0.79
b + 5m = 1.60

Subtract:
(b + 5m) - (b + 2m) = 1.60 - 0.79
→ 3m = 0.81
→ m = 0.27

Then: b + 2×0.27 = 0.79 → b + 0.54 = 0.79 → b = 0.25

For 9 minutes:
Cost = 0.25 + 9×0.27 = 0.25 + 2.43 = $2.68

Fill in blanks:
The basic cost for a phone call is $0.25 and each minute costs $0.27 extra.

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Problem 5: Boat ride cost

40 miles → $405
100 miles → $825

Let:
Basic cost = b
Cost per mile = m

Equations:
b + 40m = 405
b + 100m = 825

Subtract:
60m = 420 → m = 7

Then: b + 40×7 = 405 → b + 280 = 405 → b = 125

For 75-mile trip:
Cost = 125 + 75×7 = 125 + 525 = $650

Fill in blanks:
The basic cost for a boat ride is $125 and each kilometer costs... wait! It says “kilometer” but the problem uses miles. Probably typo — should be “mile”.

Assuming it’s “mile”:
each mile costs $7 extra.

*(If really kilometer, we’d need conversion, but no — problem uses miles throughout. Likely error in blank. We’ll go with mile.)*

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Problem 6: Production cost

4 units → $204.80
8 units → $209.60

Let:
Fixed cost = F
Cost per unit = C

Equations:
F + 4C = 204.80
F + 8C = 209.60

Subtract:
4C = 4.80 → C = 1.20

Then: F + 4×1.20 = 204.80 → F + 4.80 = 204.80 → F = 200.00

For 12 units:
Cost = 200 + 12×1.20 = 200 + 14.40 = $214.40

Fill in blanks:
The fixed cost for producing units is $200.00 and each additional unit costs $1.20 extra to produce.

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Problem 7: Container with water cups

0 cups → 0.75 lbs
3 cups → 2.25 lbs

Weight of 3 cups of water = 2.25 - 0.75 = 1.50 lbs
So 1 cup = 1.50 / 3 = 0.50 lbs

With 5 cups:
Water weight = 5 × 0.50 = 2.50 lbs
Container = 0.75 lbs
Total = 2.50 + 0.75 = 3.25 lbs

Fill in blank:
A cup of water weighs 0.50 pounds.

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Problem 8: Plane descending

After 5 min → 6500 ft
After 7 min → 5900 ft

Time between: 2 minutes
Drop: 6500 - 5900 = 600 ft
So rate = 600 / 2 = 300 ft per minute

Now, find elevation at time 0 (when descent started?):

At 5 min: 6500 ft
So in 5 min, it dropped 5 × 300 = 1500 ft
So starting elevation = 6500 + 1500 = 8000 ft

Check with 7 min: 7 × 300 = 2100 ft drop → 8000 - 2100 = 5900 ✓

A) After 10 minutes:
Drop = 10 × 300 = 3000 ft
Elevation = 8000 - 3000 = 5000 ft

B) When does it land? At sea level = 0 ft
So 8000 - 300t = 0 → 300t = 8000 → t = 8000 / 300 ≈ 26.666... minutes

That’s 26 minutes and 40 seconds, but since question likely wants decimal or fraction:

t = 80/3 ≈ 26.67 minutes (or exactly 26⅔ minutes)

But let’s write as fraction: 8000 ÷ 300 = 80/3 = 26⅔ minutes

Fill in answers:

A) 5000 ft
B) 26⅔ minutes (or 26.67 if decimal preferred)

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Now, compiling all final answers as requested.

Final Answer:

1) Fare for 2 km trip: $2.50
Blanks: basic cost = $0.90, per km = $0.80

2) Length with 90g: 7.45 cm
Blanks: original length = 5.2 cm, per gram = 0.025 cm

3) Rod length at 75°C: 72.75 cm
Blanks: per degree = 0.0025 cm, starting length = 72.6 cm

4) Toll for 9 min: $2.68
Blanks: basic cost = $0.25, per minute = $0.27

5) 75-mile boat ride: $650
Blanks: basic cost = $125, per mile = $7 *(note: problem says "kilometer" but should be "mile")*

6) Cost for 12 units: $214.40
Blanks: fixed cost = $200.00, per unit = $1.20

7) Weight with 5 cups: 3.25 pounds
Blank: cup of water = 0.50 pounds

8) A) Elevation after 10 min: 5000 ft
B) Time to land: 26⅔ minutes (or 26.67 minutes)