level 4
25) the average speed of marlons bike ride through the park can be found by dividing the distar
traveled by the time it takes him to finish the ride. he biked 6.5 miles, stopped to rest, then biked
another 4.25 miles. it took him 0.73 hours. (you must show all your work for full credit)

part a : using the rules for calculations with significant digits, first determine the total numbe
of miles marlon biked through the park?

part b using the rules for calculations with significant digits and the answer from part a, what
was marlons average speed for his ride through the park. explain your answer in full sentences.
(there should be ______ significant digits because _______)
a) 4.5 miles per hour
b) 4.52 miles per hour
c) 15 miles per hour
d) 15.3 miles per hour

for problems 26 – 28 using the dimensions of the sail shown
26) is the height of the sail a rational or irrational number?
explain.

27) is the width of the sail a rational or irrational number?
explain.

28) the length of the diagonal of the sail can be found
using the pythagorean theorem, where: ( d = sqrt{h^2 + w^2} ).
a) find the length of the diagonal.

b) is the length of the diagonal a rational or irrational number? explain.

(image of a right triangle with height 5 m, width 2 m)

Question

level 4
25) the average speed of marlons bike ride through the park can be found by dividing the distar
traveled by the time it takes him to finish the ride. he biked 6.5 miles, stopped to rest, then biked
another 4.25 miles. it took him 0.73 hours. (you must show all your work for full credit)

part a : using the rules for calculations with significant digits, first determine the total numbe
of miles marlon biked through the park?

part b using the rules for calculations with significant digits and the answer from part a, what
was marlons average speed for his ride through the park. explain your answer in full sentences.
(there should be ______ significant digits because _______)
    a) 4.5 miles per hour
    b) 4.52 miles per hour
    c) 15 miles per hour
    d) 15.3 miles per hour

for problems 26 – 28 using the dimensions of the sail shown
26) is the height of the sail a rational or irrational number?
explain.

27) is the width of the sail a rational or irrational number?
explain.

28) the length of the diagonal of the sail can be found
using the pythagorean theorem, where: ( d = sqrt{h^2 + w^2} ).
    a) find the length of the diagonal.

b) is the length of the diagonal a rational or irrational number? explain.

(image of a right triangle with height 5 m, width 2 m)

Accepted Answer

### Part A
# Explanation:
## Step1: Identify distances to add
We have two distances: 6.5 miles and 4.25 miles.
## Step2: Add the distances
To find the total distance, we add them: $ 6.5 + 4.25 = 10.75 $ miles. Now, considering significant digits, when adding, we go by the least number of decimal places. 6.5 has one decimal place, 4.25 has two. So we round to one decimal place? Wait, no, the problem says "first determine the total number of miles" (maybe just sum first, then handle sig figs? Wait, the question is "first determine the total number of miles Marlon biked" – maybe just sum: 6.5 + 4.25 = 10.75 miles. But let's check sig figs for addition: the number of decimal places in the result should match the least number of decimal places in the terms. 6.5 has 1 decimal place, 4.25 has 2. So we round to 1 decimal place? Wait, 6.5 + 4.25 = 10.75, which would round to 10.8? Wait, no, maybe the problem is just asking for the total distance before considering sig figs for the speed? Wait, the Part A says "Using the rules for calculations with significant digits, first determine the total number of miles". Wait, 6.5 has two significant digits? Wait, 6.5 has two, 4.25 has three. When adding, the decimal places: 6.5 is to the tenths place, 4.25 to hundredths. So the sum should be to the tenths place. 6.5 + 4.25 = 10.75, which rounds to 10.8? Wait, no, maybe I misread. Wait, 6.5 miles (two sig figs) and 4.25 miles (three sig figs). When adding, the result's precision is determined by the least precise measurement. 6.5 is precise to the tenths place, 4.25 to hundredths. So we round the sum to the tenths place: 10.75 rounded to tenths is 10.8? Wait, but maybe the problem is just asking for the sum, not worrying about sig figs yet? Wait, the question is "first determine the total number of miles Marlon biked through the park?" So maybe just add 6.5 + 4.25 = 10.75 miles. Then, for sig figs, when adding, the number of decimal places: 6.5 has 1, 4.25 has 2, so we keep one decimal place? So 10.8 miles? Wait, no, 6.5 + 4.25 = 10.75. If we consider significant digits for addition, the rule is that the result has the same number of decimal places as the term with the least number of decimal places. 6.5 has 1 decimal place, so the sum should have 1 decimal place. So 10.8 miles? Wait, but 6.5 is two significant digits, 4.25 is three. Maybe the problem is just asking for the total distance, so 6.5 + 4.25 = 10.75 miles. Let's proceed.

# Answer:
The total number of miles Marlon biked is $ \boldsymbol{10.75} $ miles (or $ \boldsymbol{10.8} $ miles if rounding for sig figs in addition, but likely just the sum here).

### Part B
# Explanation:
## Step1: Recall average speed formula
Average speed $ v = \frac{\text{total distance}}{\text{total time}} $. From Part A, total distance is 10.75 miles (or 10.8, but let's use 10.75 for accuracy), time is 0.73 hours.
## Step2: Calculate speed
$ v = \frac{10.75}{0.73} \approx 14.726 $ miles per hour. Wait, but the options are 4.5, 4.52, 15, 15.3. Wait, maybe I made a mistake in Part A. Wait, wait, the problem says "the average speed of Marlon's bike ride through the park can be found by dividing the distance traveled by the time it takes him to finish the ride. He biked 6.5 miles, stopped to rest, then biked another 4.25 miles. It took him 0.73 hours." Wait, maybe I added wrong? 6.5 + 4.25 is 10.75? Wait, 6 + 4 is 10, 0.5 + 0.25 is 0.75, so 10.75. Then 10.75 / 0.73 ≈ 14.72, which is close to 15 (option C) or 15.3 (D). Wait, maybe I messed up the distance. Wait, maybe the time is 0.73 hours, but let's recalculate: 10.75 ÷ 0.73. Let's do 0.73 × 15 = 10.95, which is more than 10.75. 0.73 × 14.7 = 0.73×14 + 0.73×0.7 = 10.22 + 0.511 = 10.731, which is very close to 10.75. So approximately 14.7, which rounds to 15 (two significant digits) or 15.3 (three). Wait, the time is 0.73 hours (two significant digits), the distance: 6.5 (two) and 4.25 (three). When adding, 6.5 + 4.25 = 10.75, but since 6.5 has two decimal places? No, 6.5 has one decimal place (tenths), 4.25 has two (hundredths). So the sum should have one decimal place: 10.8 (three significant digits? Wait, 10.8 has three: 1, 0, 8? No, trailing zero after decimal is significant, but 10.8 has three significant digits. Time is 0.73 (two significant digits). When dividing, the result should have the same number of significant digits as the least precise measurement. So distance (if 10.8, three sig figs) and time (two sig figs), so the speed should have two sig figs. 10.75 / 0.73 ≈ 14.726, which rounds to 15 (two sig figs). So the answer is C) 15 miles per hour. The number of significant digits is two because the time (0.73) has two significant digits, and when dividing, the result takes the least number of significant digits from the inputs.

# Answer:
There should be $\boldsymbol{2}$ significant digits because the time (0.73 hours) has 2 significant digits, and when dividing, the result’s significant digits match the least number of significant digits in the values used (distance after addition considering sig figs or time, here time has 2). The average speed is $\boldsymbol{C. 15}$ miles per hour.

### Problem 26
# Brief Explanations:
The height of the sail is given as 5 m. A rational number is a number that can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b
eq 0$. 5 can be written as $\frac{5}{1}$, so it is a rational number.
# Answer:
The height of the sail is a rational number because 5 can be expressed as $\frac{5}{1}$ (a fraction of integers), so it is rational.

### Problem 27
# Brief Explanations:
The width of the sail is given as 2 m. 2 can be written as $\frac{2}{1}$, which is a fraction of integers, so it is a rational number.
# Answer:
The width of the sail is a rational number because 2 can be expressed as $\frac{2}{1}$ (a fraction of integers), so it is rational.

### Problem 28a
# Explanation:
## Step1: Identify values for Pythagorean Theorem
The height $ h = 5 $ m, width $ w = 2 $ m. The formula for the diagonal $ d = \sqrt{h^2 + w^2} $.
## Step2: Calculate $ h^2 $ and $ w^2 $
$ h^2 = 5^2 = 25 $, $ w^2 = 2^2 = 4 $.
## Step3: Sum the squares
$ h^2 + w^2 = 25 + 4 = 29 $.
## Step4: Take the square root
$ d = \sqrt{29} \approx 5.385 $ m (or leave as $\sqrt{29}$).

# Answer:
The length of the diagonal is $ \boldsymbol{\sqrt{29} \approx 5.39} $ meters (or exactly $ \sqrt{29} $ meters).

### Problem 28b
# Brief Explanations:
The diagonal length is $ \sqrt{29} $. A rational number can be expressed as a fraction of integers, but $ \sqrt{29} $ is a square root of a non - perfect square. Since 29 is a prime number and not a perfect square, $ \sqrt{29} $ cannot be written as a fraction of integers, so it is an irrational number.
# Answer:
The length of the diagonal is an irrational number because $ \sqrt{29} $ (the diagonal length) cannot be expressed as a fraction of two integers (since 29 is not a perfect square), so it is irrational.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Part A

Step1: Identify distances to add

Step2: Add the distances

Step1: Recall average speed formula

Step2: Calculate speed

Answer:

Part B