Let's go through each question one by one:
Question 1
We have 50 students in age group 7–12, and 23 liked adventure or biography. Therefore, the number of students in the 7-12 age group who liked detective is:
- 50 - 23 = 27 students.
The total number of students who liked detective books from both age groups is 28. Thus, to find how many students in the age group 13-18 liked adventure or biography, we subtract the detective fans:
- 28 (total detective fans) - 27 (detective fans in 7–12 age group) = 1 student in the age group 13-18 liked detective.
Now, if the total in the 13-18 age group is unknown, let’s say "x" is the number of students in the 13-18 who liked adventure or biography. Assuming there are the same total study participants (100 students), we deduce that:
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Total students who liked adventure or biography in age group 13–18 = Total students - Students in 7-12 who liked adventure or biography
-
So,
x = 100 - 50 - 1 (who liked detective) = 49.
Answer: 49
Question 2
The situation describes an event where a specific action (taking medicine) leads to a specific outcome (recovering faster). Therefore, this implies a cause-and-effect relationship.
Answer: This is an example of causation.
Question 3
To find the function representing the bulldozer's value after t years, we see that it starts at $63,103 and depreciates. After year 1, the value is $58,054.76; after year 2 it is $53,410.38. To find the rate:
Value after 1 year: \( 58,054.76 = 63,103(1 - r) \) Where "r" is the depreciation rate.
Calculating for "r": \( r \approx 0.079 \) or about 8%. The function can be written as:
- \( f(t) = 63,103(0.92)^t \)
Answer: f(t) = 63,103(0.92)t
Question 4
The home value increases by 5% each year. The equation that models this is:
- \( f(x) = 240,000(1.05)^x \)
Answer: f(x) = 240000(1.05)x
Question 5
The function \( f(x) = (0.93)^x \) represents exponential decay because the base is less than 1.
Answer: Exponential decay of 7%.
Question 6
The increase in sales is a constant percentage, indicating it follows an exponential growth pattern, not a linear one.
Answer: Exponentially, because the table shows a constant percentage increase in sales per month.
Question 7
For residual analysis, a small residual suggests a good fit. Here, we see both positives and negatives across the months, indicating varying degrees of error, but it's not consistently far from zero.
Answer: Yes, the equation is a good fit because the residuals are not all far from zero.
Question 8
The y-intercept in the function represents the initial value of the dependent variable when the independent variable (sunlight hours) is zero.
Answer: The original height of the plant was 4 mm.
Question 9
The data set represents numerical values (ages) related to individuals, making it univariate.
Answer: Numerical and univariate.
Question 10
The y-intercepts can be determined based on \( f(x) \) when \( x = 0 \):
- \( f(0) = 3(1.02)^0 = 3 \)
Checking \( g(x) \):
- g(0) is given in the table as 6.
This means: Answer: The y-intercept of g(x) is equal to 2 times the y-intercept of f(x).
Question 11
Grade point average is numerical. Therefore, a numerical representation, such as a bar chart, is suitable.
Answer: Bar chart; because grade point average is numerical.
Question 12
The fraction 0.96 indicates the house's value decreases by 4% each year, which means it retains 96% of its value annually.
Answer: The decrease in the value of the house per year, which is 4%.
Question 13
To simplify \( \frac{\sqrt{2}}{\sqrt[3]{2}} \): This can be rewritten as:
- \( \frac{2^{1/2}}{2^{1/3}} = 2^{(1/2 - 1/3)} = 2^{1/6} \).
Answer: 2 to the power of one sixth.
Question 14
For \( 2^{\frac{4}{3}} \): This can be rewritten as:
- \( \sqrt[3]{2^4} = \sqrt[3]{16} \).
Answer: cube root of 16.
Question 15
For the function \( f(x) = 3(1.5)^x \), it represents an exponential function. The point where it crosses the y-axis at x = 0 gives:
- \( f(0) = 3 \).
Answer: graph of increasing exponential function going through point 0, 3.
1 answer