Answer :
To determine which situations can be modeled by the equation [tex]\( y = \frac{5}{2} x + 20 \)[/tex], we need to analyze the properties of the equation and then see if each of the given situations fits those properties.
The equation [tex]\( y = \frac{5}{2} x + 20 \)[/tex] represents a linear relationship where:
- The [tex]\( y \)[/tex]-intercept is 20, meaning that when [tex]\( x = 0 \)[/tex], [tex]\( y = 20 \)[/tex].
- The slope is [tex]\(\frac{5}{2}\)[/tex] or 2.5, which shows that for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2.5 units.
Let's analyze each situation:
A. Tonya observed a culture of 20 cells. The number of cells in the culture increases by two and a half times each day.
- Here, the initial number of cells is 20 (which matches the [tex]\( y \)[/tex]-intercept 20).
- The number of cells increasing by two and a half times each day suggests a multiplicative increase rather than a linear function. This does not match the given equation.
B. An airport shuttle service charges a \[tex]$20.00 pick-up fee plus an additional \$[/tex]2.50 per mile driven.
- The \[tex]$20.00 pick-up fee corresponds to the \( y \)-intercept 20. - The \$[/tex]2.50 per mile corresponds to the slope of 2.5.
- This fits the situation perfectly.
C. Corey has already walked 20 feet. He will continue to walk at a rate of 5 feet every 2 seconds.
- The initial 20 feet corresponds to the [tex]\( y \)[/tex]-intercept 20.
- Walking 5 feet every 2 seconds translates to a rate of [tex]\(\frac{5}{2} = 2.5\)[/tex] feet per second, matching the slope 2.5.
- This situation fits the equation.
D. Chef Paulus has baked [tex]\( \frac{21}{2} \)[/tex] dozen cookies so far. He will continue to bake 20 dozen more per day.
- The initial [tex]\( \frac{21}{2} \)[/tex] dozen (which is 10.5 dozen) does not match the intercept of 20.
- The rate of 20 dozen per day is not consistent with the slope of 2.5.
- This situation does not fit the equation.
E. The temperature is 20 degrees Fahrenheit. The temperature will decrease by [tex]\( \frac{21}{2} \)[/tex] degrees every hour.
- The initial 20 degrees matches the [tex]\( y \)[/tex]-intercept 20.
- However, the temperature decreasing by 2.5 degrees per hour means a negative slope of -2.5, not a positive 2.5.
- This situation does not fit the equation.
Based on the above analysis, the situations that could be modeled by the equation [tex]\( y = \frac{5}{2} x + 20 \)[/tex] are:
- Option B: An airport shuttle service charges a \[tex]$20.00 pick-up fee plus an additional \$[/tex]2.50 per mile driven.
- Option C: Corey has already walked 20 feet. He will continue to walk at a rate of 5 feet every 2 seconds.
So the correct answer is:
- B, C
The equation [tex]\( y = \frac{5}{2} x + 20 \)[/tex] represents a linear relationship where:
- The [tex]\( y \)[/tex]-intercept is 20, meaning that when [tex]\( x = 0 \)[/tex], [tex]\( y = 20 \)[/tex].
- The slope is [tex]\(\frac{5}{2}\)[/tex] or 2.5, which shows that for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2.5 units.
Let's analyze each situation:
A. Tonya observed a culture of 20 cells. The number of cells in the culture increases by two and a half times each day.
- Here, the initial number of cells is 20 (which matches the [tex]\( y \)[/tex]-intercept 20).
- The number of cells increasing by two and a half times each day suggests a multiplicative increase rather than a linear function. This does not match the given equation.
B. An airport shuttle service charges a \[tex]$20.00 pick-up fee plus an additional \$[/tex]2.50 per mile driven.
- The \[tex]$20.00 pick-up fee corresponds to the \( y \)-intercept 20. - The \$[/tex]2.50 per mile corresponds to the slope of 2.5.
- This fits the situation perfectly.
C. Corey has already walked 20 feet. He will continue to walk at a rate of 5 feet every 2 seconds.
- The initial 20 feet corresponds to the [tex]\( y \)[/tex]-intercept 20.
- Walking 5 feet every 2 seconds translates to a rate of [tex]\(\frac{5}{2} = 2.5\)[/tex] feet per second, matching the slope 2.5.
- This situation fits the equation.
D. Chef Paulus has baked [tex]\( \frac{21}{2} \)[/tex] dozen cookies so far. He will continue to bake 20 dozen more per day.
- The initial [tex]\( \frac{21}{2} \)[/tex] dozen (which is 10.5 dozen) does not match the intercept of 20.
- The rate of 20 dozen per day is not consistent with the slope of 2.5.
- This situation does not fit the equation.
E. The temperature is 20 degrees Fahrenheit. The temperature will decrease by [tex]\( \frac{21}{2} \)[/tex] degrees every hour.
- The initial 20 degrees matches the [tex]\( y \)[/tex]-intercept 20.
- However, the temperature decreasing by 2.5 degrees per hour means a negative slope of -2.5, not a positive 2.5.
- This situation does not fit the equation.
Based on the above analysis, the situations that could be modeled by the equation [tex]\( y = \frac{5}{2} x + 20 \)[/tex] are:
- Option B: An airport shuttle service charges a \[tex]$20.00 pick-up fee plus an additional \$[/tex]2.50 per mile driven.
- Option C: Corey has already walked 20 feet. He will continue to walk at a rate of 5 feet every 2 seconds.
So the correct answer is:
- B, C
