Athar is buying hats for his baseball team. The shipping charge is $5, and each hat costs $8. There are ten people on his baseball team. The graph shows the total cost of the hats.... Athar is buying hats for his baseball team. The shipping charge is $5, and each hat costs $8. There are ten people on his baseball team. The graph shows the total cost of the hats. Select all true statements. A) The ordered pair for 1 hat bought is (1, 8). B) The domain is whole numbers of hats from 1 to 10. C) The range is numbers of dollars from 0 to 100 in increments of 5. D) The ordered pair for 5 hats bought is (5, 45). E) The cost of ordering 10 hats is double the cost of ordering 5 hats.
Understand the Problem
The question is asking about the graph that represents the total cost of buying hats, including details such as ordered pairs, domain, range, and relationships between the costs of different quantities of hats. We need to validate statements based on this information.
Answer
B, C, D are true statements.
Answer for screen readers
The true statements are:
- A: False (ordered pair is (1, 13))
- B: True (domain is whole numbers from 0 to 10)
- C: True (range is from $5$ to $85$, in increments of $8$)
- D: True (ordered pair for 5 hats is (5, 45))
- E: False (cost of ordering 10 hats is not double the cost of ordering 5 hats)
Steps to Solve
- Understand the Cost Equation
The total cost $C$ of buying $h$ hats can be determined with this formula: $$ C = 8h + 5 $$ Here, $8$ is the cost of each hat, and $5$ is the shipping charge.
- Calculate Cost for 1 Hat
For $h = 1$: $$ C = 8(1) + 5 = 8 + 5 = 13 $$ The ordered pair for 1 hat is $(1, 13)$.
- Check the Domain
The domain represents the number of hats that can be bought, which will be whole numbers from $0$ to $10$ (the number of players). Thus, the domain is: $$ {0, 1, 2, \ldots, 10} $$
- Determine the Range
Next, we calculate the costs for buying up to $10$ hats. The highest cost for $h = 10$: $$ C = 8(10) + 5 = 80 + 5 = 85 $$ So the range will be: $$ {5, 13, 21, \ldots, 85} $$ This shows that the costs go from $5$ to $85$, in increments of $8$ after the shipping charge.
- Assess Cost for 5 Hats and 10 Hats
Calculate the ordered pair for $h = 5$: $$ C = 8(5) + 5 = 40 + 5 = 45 $$ The ordered pair is $(5, 45)$.
Then for $h = 10$: $$ C = 8(10) + 5 = 85 $$ This means the cost for 10 hats ($85$) is not double the cost for 5 hats ($45$), since $85$ is not equal to $90$.
The true statements are:
- A: False (ordered pair is (1, 13))
- B: True (domain is whole numbers from 0 to 10)
- C: True (range is from $5$ to $85$, in increments of $8$)
- D: True (ordered pair for 5 hats is (5, 45))
- E: False (cost of ordering 10 hats is not double the cost of ordering 5 hats)
More Information
The graph represents the total cost for purchasing hats, which includes the fixed shipping charge. Each ordered pair corresponds to the number of hats bought and the total cost calculated via the defined formula.
Tips
- Miscalculating the total cost by not adding the shipping charge correctly.
- Confusing the ordered pairs with costs at different quantities.
- Overlooking the definition of domain and range, particularly with respect to whole numbers.
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