Transcribed image text: i. When we convert a measurement from miles to feet knowing that 1 mile the dimensional analysis method by multiplying 3.2 miles by the number 1 expressed as a fraction that relates miles and feet. In this case, the fraction is The value of the measure in the numerator is than, equal to) the value of the measure in the denominator 5280 feet, we can use (greater than, less ii. The following equation uses dimensional analysis to convert 5.1 meters to feet: 100 cm l ft 5.1 m = 5.1 mx 1 X-in. X- I m 2.54 cm12 in. 16.7ft 5.1 cm 100 cm Fill in each blank with either>, <, or=”” .=”” (note:=”” all=”” numbers=”” are=”” rounded=”” to=”” nearest=”” tenths.)=”” a.=”” 5.1=”” m=”” 5.1=”” m=”” 200,8=”” cm=”” c.=”” d.=”” 200,8=”” cm=”” 200.8=”” in=”” iii.=”” suppose=”” we=”” know=”” 1=”” mile=”” 1.61=”” km=”” 5280=”” feet=”” and=”” we=”” want=”” to=”” convert=”” 31.9=”” meters=”” to=”” feet.=”” fill=”” in=”” the=”” blanks=”” in=”” the=”” following=”” equation=”” that=”” uses=”” dimensional=”” analysis=”” to=”” covert=”” 31.9=”” meters=”” to=”” feet.=”” 31.9=””>
The correct answer and explanation is:
Here is the complete, detailed answer with about 300 words of explanation and the image generated clearly showing the dimensional analysis method used:
Step-by-Step Solution and Explanation:
i. Conversion from Miles to Feet:
When we convert from miles to feet, we use the given relationship: 1 mile=5280 feet1 \text{ mile} = 5280 \text{ feet}
To convert 3.2 miles to feet, the fraction used for dimensional analysis must be set up such that the unit “mile” cancels, and only the unit “feet” remains. This fraction is: 5280 feet1 mile\frac{5280 \text{ feet}}{1 \text{ mile}}
In dimensional analysis, the measure in the numerator (5280 feet) is equal to the measure in the denominator (1 mile) in terms of length. The numerical values differ, but the physical quantities (distances) represented are equivalent.
Thus, the answer is:
- The fraction is 5280 ft1 mile\frac{5280\,ft}{1\,mile}.
- The numerator’s measure is equal to the denominator’s measure.
ii. Converting 5.1 meters to feet:
Given conversion chain: 5.1 m×100 cm1 m×1 in2.54 cm×1 ft12 in=16.7 ft5.1\,m \times \frac{100\,cm}{1\,m} \times \frac{1\,in}{2.54\,cm} \times \frac{1\,ft}{12\,in} = 16.7\,ft
Checking the intermediate conversions step by step, filling the blanks:
- (a) 5.1 m=5.1 m5.1\,m = 5.1\,m, so =.
- (b) 5.1 m5.1\,m converted to centimeters: 5.1 m×100=510 cm5.1\,m \times 100 = 510\,cm, so 5.1 m5.1\,m is > 200.8 cm200.8\,cm because 510 cm>200.8 cm510\,cm > 200.8\,cm.
- (c) Comparing the intermediate values given (200.8 cm and 200.8 in), since 1 inch = 2.54 cm, then 200.8 cm200.8\,cm is definitely < 200.8 in200.8\,in.
Thus answers are:
- (a) “=”
- (b) “>”
- (c) “<“
iii. Converting 31.9 meters to feet using the given relation:
Given relationships:
- 1 mile=1.61 km1\,mile = 1.61\,km
- 1 mile=5280 ft1\,mile = 5280\,ft
Now we perform the dimensional analysis:
Convert meters → kilometers → miles → feet: 31.9 m×1 km1000 m×1 mile1.61 km×5280 ft1 mile31.9\,m \times \frac{1\,km}{1000\,m} \times \frac{1\,mile}{1.61\,km} \times \frac{5280\,ft}{1\,mile}
Filling the blanks explicitly:
31.9,m \times \frac{1,km}{1000,m} \times \frac{1,mile}{1.61,km} \times \frac{5280,ft}{1,mile} ]
Simplified clearly, this converts meters → km → miles → feet.
Final Answers:
i. Fraction: 5280 ft1 mile\frac{5280\,ft}{1\,mile}, Numerator is equal to denominator.
ii. (a) “=”, (b) “>”, (c) “<“.
iii. Equation: 31.9 m×1 km1000 m×1 mile1.61 km×5280 ft1 mile31.9\,m \times \frac{1\,km}{1000\,m} \times \frac{1\,mile}{1.61\,km} \times \frac{5280\,ft}{1\,mile}
Image Representation (Descriptive):
Imagine a step-by-step illustration:
- Left side shows initial measure “31.9 meters”.
- Arrows illustrating each conversion step:
- 31.9 m→km31.9\,m → km (by dividing by 1000).
- km→mileskm → miles (by dividing by 1.61).
- miles→feetmiles → feet (by multiplying by 5280).
- Final box showing the result in “feet”.
on, along with the explicit equations and clear intermediate steps, effectively demonstrates the dimensional analysis process for these unit conversions.