complete the following scientific calculations and make sure to keep track of units and sig figs!
1) 3.24 in x 2.3 in
6) 7.89 m − 0.003782 m
2) 1.543 m x 4.32 m
7) 345.06 kg + 1.6 kg
3) 1.34 cm x 1.5 cm x 2.35 cm
8) 575 kg x 0.000458 m ÷ 4.423 s ÷ 0.0389 s
4) 82.5 g ÷ 35.2 ml
9) 143 g ÷ 1.24 cm ÷ 2.39 cm ÷ 5.35 cm
5) 120 ft − 93.4 ft + 81.2 ft
10) 93000000 miles ÷ 8766 hours

Question

complete the following scientific calculations and make sure to keep track of units and sig figs!
1)  3.24 in x 2.3 in
6)  7.89 m − 0.003782 m
2)  1.543 m x 4.32 m
7)  345.06 kg + 1.6 kg
3)  1.34 cm x 1.5 cm x 2.35 cm
8)  575 kg x 0.000458 m ÷ 4.423 s ÷ 0.0389 s
4)  82.5 g ÷ 35.2 ml
9)  143 g ÷ 1.24 cm ÷ 2.39 cm ÷ 5.35 cm
5)  120 ft − 93.4 ft + 81.2 ft
10) 93000000 miles ÷ 8766 hours

Accepted Answer

### Problem 1: $ 3.24 \, \text{in} \times 2.3 \, \text{in} $
# Explanation:
## Step 1: Multiply the numbers
Multiply $ 3.24 $ and $ 2.3 $. $ 3.24 \times 2.3 = 7.452 $
## Step 2: Determine significant figures
For multiplication, the result should have the same number of significant figures as the least precise measurement. $ 3.24 $ has 3 sig figs, $ 2.3 $ has 2 sig figs. So we round to 2 sig figs.
$ 7.452 \approx 7.5 $ (rounded to 2 sig figs)
## Step 3: Include units
The units are square inches ($ \text{in}^2 $) since we multiplied inches by inches.
# Answer: $ 7.5 \, \text{in}^2 $

### Problem 2: $ 1.543 \, \text{m} \times 4.32 \, \text{m} $
# Explanation:
## Step 1: Multiply the numbers
$ 1.543 \times 4.32 = 6.66576 $
## Step 2: Determine significant figures
$ 1.543 $ has 4 sig figs, $ 4.32 $ has 3 sig figs. Round to 3 sig figs.
$ 6.66576 \approx 6.67 $
## Step 3: Include units
Units are square meters ($ \text{m}^2 $).
# Answer: $ 6.67 \, \text{m}^2 $

### Problem 3: $ 1.34 \, \text{cm} \times 1.5 \, \text{cm} \times 2.35 \, \text{cm} $
# Explanation:
## Step 1: Multiply the numbers
First, $ 1.34 \times 1.5 = 2.01 $, then $ 2.01 \times 2.35 = 4.7235 $
## Step 2: Determine significant figures
$ 1.34 $ (3 sig figs), $ 1.5 $ (2 sig figs), $ 2.35 $ (3 sig figs). Least is 2 sig figs? Wait, no: in multiplication, the number of sig figs is determined by the least number. Wait, $ 1.5 $ has 2, but wait, $ 1.5 $ could be considered as 2 or maybe 1? Wait, no, $ 1.5 $ has two significant figures. Wait, but $ 1.34 $ (3), $ 1.5 $ (2), $ 2.35 $ (3). So the result should have 2 sig figs? Wait, no, wait: $ 1.5 $ is two, so when multiplying, the result should have two? Wait, no, let's check again. Wait, $ 1.34 \times 1.5 = 2.01 $ (but $ 1.5 $ has two sig figs, so $ 2.01 $ should be rounded to 2.0? Wait, no, maybe I made a mistake. Wait, the rule is that for multiplication/division, the result has the same number of sig figs as the input with the least number of sig figs. So $ 1.34 $ (3), $ 1.5 $ (2), $ 2.35 $ (3). So the least is 2. So after multiplying all three, we round to 2 sig figs. Wait, but let's do the multiplication first: $ 1.34 \times 1.5 = 2.01 $, then $ 2.01 \times 2.35 = 4.7235 $. Now, round to 2 sig figs: $ 4.7235 \approx 4.7 $? Wait, no, 4.7235 rounded to two sig figs is 4.7? Wait, no, 4.7235: the first two sig figs are 4 and 7, the next digit is 2, which is less than 5, so we keep it 4.7? Wait, no, 4.7235: 4.7 (two sig figs). Wait, but maybe I messed up the sig figs for $ 1.5 $. Wait, $ 1.5 $ has two significant figures. So the product should have two. So $ 4.7235 \approx 4.7 \, \text{cm}^3 $? Wait, no, wait, $ 1.34 $ is three, $ 1.5 $ is two, $ 2.35 $ is three. So the limiting factor is two. So the answer should have two sig figs. So $ 4.7235 \approx 4.7 \, \text{cm}^3 $? Wait, but let's check the multiplication again. Wait, $ 1.34 \times 1.5 = 2.01 $ (but with two sig figs, that would be 2.0), then $ 2.0 \times 2.35 = 4.7 $. Yes, that's correct. So the result is $ 4.7 \, \text{cm}^3 $.
## Step 3: Include units
Units are cubic centimeters ($ \text{cm}^3 $).
# Answer: $ 4.7 \, \text{cm}^3 $ (or maybe 4.72 if we consider $ 1.5 $ as two sig figs but maybe the problem expects three? Wait, no, $ 1.5 $ has two. So 4.7 is correct.)

### Problem 4: $ 82.5 \, \text{g} \div 35.2 \, \text{mL} $
# Explanation:
## Step 1: Divide the numbers
$ 82.5 \div 35.2 \approx 2.34375 $
## Step 2: Determine significant figures
Both $ 82.5 $ (3 sig figs) and $ 35.2 $ (3 sig figs) have 3 sig figs. So round to 3 sig figs.
$ 2.34375 \approx 2.34 $
## Step 3: Include units
Units are grams per milliliter ($ \text{g/mL} $).
# Answer: $ 2.34 \, \text{g/mL} $

### Problem 5: $ 120 \, \text{ft} - 93.4 \, \text{ft} + 81.2 \, \text{ft} $
# Explanation:
## Step 1: Perform subtraction first
$ 120 - 93.4 = 26.6 $
## Step 2: Perform addition
$ 26.6 + 81.2 = 107.8 $
## Step 3: Determine significant figures
For addition/subtraction, the result should have the same number of decimal places as the least precise measurement. $ 120 $ has 0 decimal places, $ 93.4 $ has 1, $ 81.2 $ has 1. So we round to 0 decimal places? Wait, no: $ 120 $ could be considered as having one significant figure? Wait, no, $ 120 $ – if it's written as 120 without a decimal, it's ambiguous, but often in such problems, trailing zeros without a decimal are not significant. Wait, but maybe the problem considers $ 120 $ as having two significant figures (1 and 2), and the others have three. Wait, no, for addition/subtraction, the rule is about decimal places. $ 120 $ has 0 decimal places, $ 93.4 $ has 1, $ 81.2 $ has 1. So the result should have 0 decimal places? Wait, but $ 120 - 93.4 = 26.6 $ (which has 1 decimal place), then $ 26.6 + 81.2 = 107.8 $. Now, $ 120 $ has 0 decimal places, so we round $ 107.8 $ to 0 decimal places: $ 108 \, \text{ft} $. Wait, but maybe $ 120 $ is considered as having three significant figures (if it's written as $ 120. $), but it's not. So maybe the problem expects us to consider the decimal places. $ 120 $ (0 decimal), $ 93.4 $ (1), $ 81.2 $ (1). So the sum should have 0 decimal places. So $ 107.8 \approx 108 \, \text{ft} $.
# Answer: $ 108 \, \text{ft} $ (or 107.8 if we consider $ 120 $ as having one decimal place, but that's unlikely)

### Problem 6: $ 7.89 \, \text{m} - 0.003782 \, \text{m} $
# Explanation:
## Step 1: Subtract the numbers
$ 7.89 - 0.003782 = 7.886218 $
## Step 2: Determine significant figures
For subtraction, the result should have the same number of decimal places as the least precise measurement. $ 7.89 $ has 2 decimal places, $ 0.003782 $ has 6. So we round to 2 decimal places.
$ 7.886218 \approx 7.89 \, \text{m} $ (Wait, 7.886218 rounded to two decimal places is 7.89? Wait, 7.886218: the third decimal is 6, which is more than 5, so we round up the second decimal: 88 becomes 89. So 7.89 m.
# Answer: $ 7.89 \, \text{m} $

### Problem 7: $ 345.06 \, \text{kg} + 1.6 \, \text{kg} $
# Explanation:
## Step 1: Add the numbers
$ 345.06 + 1.6 = 346.66 $
## Step 2: Determine significant figures
For addition, the result should have the same number of decimal places as the least precise measurement. $ 345.06 $ has 2 decimal places, $ 1.6 $ has 1. So we round to 1 decimal place.
$ 346.66 \approx 346.7 \, \text{kg} $
# Answer: $ 346.7 \, \text{kg} $

### Problem 8: $ 575 \, \text{kg} \times 0.000458 \, \text{m} \div 4.423 \, \text{s} \div 0.0389 \, \text{s} $
# Explanation:
## Step 1: Multiply the first two numbers
$ 575 \times 0.000458 = 0.26335 $
## Step 2: Divide by the third number
$ 0.26335 \div 4.423 \approx 0.05954 $
## Step 3: Divide by the fourth number
$ 0.05954 \div 0.0389 \approx 1.5306 $
## Step 4: Determine significant figures
$ 575 $ (3 sig figs), $ 0.000458 $ (3 sig figs), $ 4.423 $ (4 sig figs), $ 0.0389 $ (3 sig figs). The least number of sig figs is 3. So round to 3 sig figs.
$ 1.5306 \approx 1.53 $
## Step 5: Include units
Units: $ \text{kg} \times \text{m} \div \text{s} \div \text{s} = \text{kg·m/s}^2 $ (which is Newtons, but we can just write the combined units)
# Answer: $ 1.53 \, \text{kg·m/s}^2 $ (or $ 1.53 \, \text{N} $)

### Problem 9: $ 143 \, \text{g} \div 1.24 \, \text{cm} \div 2.39 \, \text{cm} \div 5.35 \, \text{cm} $
# Explanation:
## Step 1: Divide $ 143 $ by $ 1.24 $
$ 143 \div 1.24 \approx 115.3226 $
## Step 2: Divide by $ 2.39 $
$ 115.3226 \div 2.39 \approx 48.2521 $
## Step 3: Divide by $ 5.35 $
$ 48.2521 \div 5.35 \approx 9.0191 $
## Step 4: Determine significant figures
$ 143 $ (3 sig figs), $ 1.24 $ (3), $ 2.39 $ (3), $ 5.35 $ (3). So round to 3 sig figs.
$ 9.0191 \approx 9.02 $
## Step 5: Include units
Units: $ \text{g} \div \text{cm} \div \text{cm} \div \text{cm} = \text{g/cm}^3 $ (density)
# Answer: $ 9.02 \, \text{g/cm}^3 $

### Problem 10: $ 93000000 \, \text{miles} \div 8766 \, \text{hours} $
# Explanation:
## Step 1: Divide the numbers
$ 93000000 \div 8766 \approx 10600.0456 $ (Wait, let's calculate it properly: $ 93000000 \div 8766 \approx 10600 $ (approximate). Wait, more accurately: $ 93000000 \div 8766 \approx 10600.0456 $)
## Step 2: Determine significant figures
$ 93000000 $ – trailing zeros without a decimal: let's assume it has two significant figures (9 and 3), and $ 8766 $ has four. So the result should have two significant figures.
$ 10600.0456 \approx 1.1 \times 10^4 $ miles per hour (or $ 11000 $ miles per hour, but in scientific notation, $ 1.1 \times 10^4 $ mph)
Wait, let's check the division again: $ 93000000 \div 8766 \approx 10600 $ (when rounded to two significant figures, that's $ 1.1 \times 10^4 $ or $ 11000 $).
# Answer: $ 1.1 \times 10^4 \, \text{mph} $ (or $ 11000 \, \text{mph} $)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1: \( 3.24 \, \text{in} \times 2.3 \, \text{in} \)

Step 1: Multiply the numbers

Step 2: Determine significant figures

Step 3: Include units

Step 1: Multiply the numbers

Step 2: Determine significant figures

Step 3: Include units

Step 1: Multiply the numbers

Step 2: Determine significant figures

Step 3: Include units

Answer:

Problem 2: \( 1.543 \, \text{m} \times 4.32 \, \text{m} \)