General 1/2 Questions
Educational · TutoringQuestions
1. A ride-sharing service charges a base fee of $3 plus an additional $1.5 per kilometer traveled. What is the total cost, in dollars, of a 15-kilometer trip?
2. A bakery sells a total of $x$ loaves of bread at $\$2.50$ each and $y$ pastries at $\$4.00$ each. Write a formula to represent the total cost, $C$, of the bread and pastries sold, and then calculate the total cost if the bakery sells 30 loaves of bread and 20 pastries.
3. A pet store sells canaries for $20 and cockatiels for $50. If the store sells 8! x !8 canaries and 8! y !8 cockatiels, write an equation to represent the total amount of money, 8! M !8, made by the store. Then, find the total amount of money made if the store sells 15 canaries and 32 cockatiels.
4. Given an arithmetic sequence with the first term 8!a = -3!8 and the second term 8!b = -6!8, find the value of the 50th term, 8!t_{50}!8, and the difference between 8!t_{50}!8 and the first term.
5. Find the values of 8! a !8 and 8! d !8 for the arithmetic sequences: 9!3, 9, 15, 21, \ldots !9 9!20, 12, 4, -4, \ldots !9
Answers and Explanations
Answer 1:
9!F = 3 + 1.5 \times 15 = 3 + 22.5 = 25.5 !9 $25.50
The total cost is calculated by multiplying the distance traveled by the per-kilometer rate and adding the base fee.
Answer 2:
The formula to represent the total cost is: 8!C = 2.50x + 4.00y!8
The total cost of selling 30 loaves of bread and 20 pastries is: 9!C = 2.50(30) + 4.00(20)!9 9!C = 75 + 80!9 9!C = 155!9
The formula is derived by multiplying the cost of each item by the number of items sold and adding the results together. The total cost is then calculated by substituting the given values into the formula.
Answer 3:
The equation to represent the total amount of money made is: 9!M = 20x + 50y !9 Substituting 8! x = 15 !8 and 8! y = 32 !8, we get: 9!M = 20(15) + 50(32) !9 9!M = 300 + 1600 !9 9!M = 1900 !9
The equation is derived by multiplying the number of each type of bird sold by its price and adding the results. Substituting the given values for 8! x !8 and 8! y !8 allows us to calculate the total amount of money made.
Answer 4:
Step 1: Calculate the common difference, D
The common difference is calculated as 8!D = b - a = -6 - (-3) = -3!8.
Step 2: Calculate the difference between t50 and the first term
To find the difference between 8!t_{50}!8 and the first term, we multiply the common difference by the number of terms minus one: 8!(-3) \times (50 - 1) = (-3) \times 49 = -147!8.
Step 3: Calculate t50
Now, we add the difference to the first term to find 8!t_{50}!8: 8!t_{50} = a + (-147) = -3 + (-147) = -150!8.
The final answer is: $-150$
We used the formula for the nth term of an arithmetic sequence, 8!t_n = a + (n-1)D!8, where 8!a!8 is the first term, 8!n!8 is the term number, and 8!D!8 is the common difference.
Answer 5:
For the sequence 8! 3, 9, 15, 21, \ldots !8: 8! a = 3 !8, 8! d = 6 !8
For the sequence 8! 20, 12, 4, -4, \ldots !8: 8! a = 20 !8, 8! d = -8 !8
To find 8! a !8, identify the first term in each sequence. To find 8! d !8, subtract any term from its preceding term.