a store sells two candles. one is 14 cm tall and burns at a constant rate of 1 cm per hour, and the second candle burns according to the function $h(t) = -0.5t + 12$, where $t$ is how long the candle burns, in hours, and $h(t)$ is the height of the candle. determine and compare the domains of the candles.

the first candle has a domain of $0 \le t \le 24$, and the second candle has a domain of $0 \le t \le 14$. the second candle has a larger domain.

the first candle has a domain of $0 \le t \le 24$, and the second candle has a domain of $0 \le t \le 14$. the first candle has a larger domain.

the first candle has a domain of $0 \le t \le 14$, and the second candle has a domain of $0 \le t \le 24$. the first candle has a larger domain.

the first candle has a domain of $0 \le t \le 14$, and the second candle has a domain of $0 \le t \le 24$. the second candle has a larger domain.

Question

a store sells two candles. one is 14 cm tall and burns at a constant rate of 1 cm per hour, and the second candle burns according to the function $h(t) = -0.5t + 12$, where $t$ is how long the candle burns, in hours, and $h(t)$ is the height of the candle. determine and compare the domains of the candles.

the first candle has a domain of $0 \le t \le 24$, and the second candle has a domain of $0 \le t \le 14$. the second candle has a larger domain.

the first candle has a domain of $0 \le t \le 24$, and the second candle has a domain of $0 \le t \le 14$. the first candle has a larger domain.

the first candle has a domain of $0 \le t \le 14$, and the second candle has a domain of $0 \le t \le 24$. the first candle has a larger domain.

the first candle has a domain of $0 \le t \le 14$, and the second candle has a domain of $0 \le t \le 24$. the second candle has a larger domain.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Rate of Change", "Linear Functions", "Finding Intercepts" ], "new_concepts": [ "Domain of Linear Functions", "Real-world Domain Constraints" ], "current_concepts": [ "Rate of Change", "Linear Functions", "Finding Intercepts", "Domain of Linear Functions", "Real-world Domain Constraints" ] } ## Determine the domain of the first candle $$ \begin{aligned} &h_1(0) = 14\ &\text{Rate of change} = -1\text{ cm/hour}\ &h_1(t) = -t + 14\ &h_1(t) \ge 0 \implies -t + 14 \ge 0 \implies t \le 14\ &\text{Domain}_1: 0 \le t \le 14 \end{aligned} $$ ## Determine the domain of the second candle $$ \begin{aligned} &h(t) = -0.5t + 12\ &h(t) \ge 0 \implies -0.5t + 12 \ge 0 \implies 0.5t \le 12 \implies t \le 24\ &\text{Domain}_2: 0 \le t \le 24 \end{aligned} $$ ## Compare the domains $$ \begin{aligned} &\text{Domain}_1: [0, 14]\ &\text{Domain}_2: [0, 24]\ &24 > 14 \implies \text{The second candle has a larger domain.} \end{aligned} $$

The first candle has a domain of $0 \le t \le 24$, and the second candle has a domain of $0 \le t \le 14$. The second candle has a larger domain.

The first candle has a domain of $0 \le t \le 24$, and the second candle has a domain of $0 \le t \le 14$. The first candle has a larger domain.

The first candle has a domain of $0 \le t \le 14$, and the second candle has a domain of $0 \le t \le 24$. The first candle has a larger domain.

The first candle has a domain of $0 \le t \le 14$, and the second candle has a domain of $0 \le t \le 24$. The second candle has a larger domain.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Domain of Linear Functions" ] }

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Question

Explanation:

Determine the domain of the first candle

Determine the domain of the second candle

Compare the domains

Answer: