18. the resistance, $r$, of a material depends on the temperature and can be described by a linear equation. if $t$ is the temperature in degrees celsius, write the linear equation for a certain material if $r = 9.51$ when $t = 0$ and $r = 12.45$ when $t = 15$.

19. carlos measures the resistance of a certain material at $6.981\ \omega$ when the temperature is $70^\circ\text{f}$. later that day, the temperature has risen to $87^\circ\text{f}$ and the resistance is $7.644\ \omega$. determine the linear equation for the resistance, $r$, of this material given the temperature, $t$, in degrees fahrenheit.

Question

18. the resistance, $r$, of a material depends on the temperature and can be described by a linear equation. if $t$ is the temperature in degrees celsius, write the linear equation for a certain material if $r = 9.51$ when $t = 0$ and $r = 12.45$ when $t = 15$.

19. carlos measures the resistance of a certain material at $6.981\ \omega$ when the temperature is $70^\circ\text{f}$. later that day, the temperature has risen to $87^\circ\text{f}$ and the resistance is $7.644\ \omega$. determine the linear equation for the resistance, $r$, of this material given the temperature, $t$, in degrees fahrenheit.

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Slope Formula", "Slope-Intercept Form", "Linear Modeling" ], "new_concepts": [ "Temperature Resistance Modeling" ], "current_concepts": [ "Slope Formula", "Slope-Intercept Form", "Linear Modeling", "Temperature Resistance Modeling" ] } ## Identify the given data points for Question 18 Using the *Linear Modeling* knowledge point $$ \begin{aligned} &\text{Let } R = mT + b\ &(T_1, R_1) = (0, 9.51)\ &(T_2, R_2) = (15, 12.45) \end{aligned} $$ ## Calculate the slope and equation for Question 18 Using the *Slope Formula* and *Slope-Intercept Form* knowledge points $$ \begin{aligned} &m = \frac{12.45 - 9.51}{15 - 0} = \frac{2.94}{15} = 0.196\ &b = 9.51\ &R = 0.196T + 9.51 \end{aligned} $$ ## Identify the given data points for Question 19 Using the *Linear Modeling* knowledge point $$ \begin{aligned} &\text{Let } R = mT + b\ &(T_1, R_1) = (70, 6.981)\ &(T_2, R_2) = (87, 7.644) \end{aligned} $$ ## Calculate the slope and equation for Question 19 Using the *Slope Formula* and *Slope-Intercept Form* knowledge points $$ \begin{aligned} &m = \frac{7.644 - 6.981}{87 - 70} = \frac{0.663}{17} = 0.039\ &R - 6.981 = 0.039(T - 70)\ &R = 0.039T - 2.73 + 6.981\ &R = 0.039T + 4.251 \end{aligned} $$ ## Question 18 $R = 0.196T + 9.51$ ## Question 19 $R = 0.039T + 4.251$ { "elements": [ { "type": "functiongraph", "params": [ { "js": "0.196*x + 9.51", "latex": "R = 0.196T + 9.51" }, -2, 20 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "R = 0.196T + 9.51", "withLabel": true } }, { "type": "point", "params": [ [0, 9.51] ], "properties": { "name": "(0, 9.51)", "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [ [15, 12.45] ], "properties": { "name": "(15, 12.45)", "color": "#8C55F2", "withLabel": true } } ], "timestamps": [0.5, 1.0] } { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Linear Modeling" ] }

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Question

Explanation:

Identify the given data points for Question 18

Calculate the slope and equation for Question 18

Identify the given data points for Question 19

Calculate the slope and equation for Question 19

Answer:

Question 18

Question 19