Question

for questions 7 - 10, write an algebraic function rule that can be used to model each scenario. then explain what the slope and the vertical intercept mean in the context of the problem.

1. at a local pumpkin patch you can pick your own pumpkins. there is a $5 charge to enter the patch and then a $0.25 charge per pound of pumpkin. let p represent the weight of the pumpkin you choose and c(p) represent the total cost of the pumpkin in terms of the weight of the pumpkin.
2. you are skiing down a 1,350 meter ski slope at 60 meters per second. let t represent your skiing time in seconds and d(t) represent the distance from the bottom of the ski slope.
3. the basement of a building is 40 feet below ground level. the building’s elevator rises at a rate of 5 feet per second. you enter the elevator in the basement. let t represent the number of seconds you are in the elevator and h(t) represent how high the elevator has risen (in feet) in t seconds.
4. there is a stack of old algebra books in the library. each book is 1 1/4 inches thick. let b represent the number of books in a stack and s(b) represent the total height of the stack in inches.

Explanation:

Question 7:

• Step1: Identify the form of the linear - function

The general form of a linear function is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. In the context of cost $C(p)$ as a function of pumpkin weight $p$, the fixed cost to enter the patch is the y - intercept and the cost per pound is the slope.
$C(p)=0.25p + 5$

• Step2: Interpret the slope

The slope $m = 0.25$. It means that for each additional pound of pumpkin, the total cost increases by $0.25$ dollars.

• Step3: Interpret the vertical - intercept

The vertical - intercept $b = 5$. It represents the fixed cost of $5$ dollars to enter the pumpkin patch, regardless of the weight of the pumpkin.

Question 8:

• Step1: Identify the form of the linear - function

The initial distance from the bottom of the ski slope is $1350$ meters and the skier is moving towards the bottom at a rate of $60$ meters per second. The function $d(t)$ has the form $d(t)=1350-60t$.

• Step2: Interpret the slope

The slope $m=-60$. The negative sign indicates that the distance from the bottom of the slope is decreasing. It means that the skier is approaching the bottom of the slope at a rate of $60$ meters per second.

• Step3: Interpret the vertical - intercept

The vertical - intercept $b = 1350$. It represents the initial distance (in meters) from the bottom of the ski slope when $t = 0$ seconds.

Question 9:

• Step1: Identify the form of the linear - function

The elevator starts $40$ feet below ground level and rises at a rate of $5$ feet per second. The function $h(t)$ has the form $h(t)=5t-40$.

• Step2: Interpret the slope

The slope $m = 5$. It means that the elevator rises $5$ feet every second.

• Step3: Interpret the vertical - intercept

The vertical - intercept $b=-40$. It represents the initial position of the elevator, which is $40$ feet below ground level.

Question 10:

• Step1: Identify the form of the linear - function

Each book is $1\frac{1}{4}=\frac{5}{4}$ inches thick. If $b$ is the number of books, then the function $s(b)$ is $s(b)=\frac{5}{4}b$.

• Step2: Interpret the slope

The slope $m=\frac{5}{4}$. It means that for each additional book in the stack, the height of the stack increases by $\frac{5}{4}$ inches.

• Step3: Interpret the vertical - intercept

The vertical - intercept $b = 0$. When there are no books ($b = 0$), the height of the stack is $0$ inches.

Answer:

Question 7:

• Function: $C(p)=0.25p + 5$
• Slope: Represents the cost per pound of pumpkin ($0.25$ dollars per pound)
• Vertical - intercept: Represents the fixed cost to enter the pumpkin patch ($5$ dollars)

Question 8:

• Function: $d(t)=1350-60t$
• Slope: Represents the rate at which the skier is approaching the bottom of the slope ($- 60$ meters per second)
• Vertical - intercept: Represents the initial distance from the bottom of the ski slope ($1350$ meters)

Question 9:

• Function: $h(t)=5t-40$
• Slope: Represents the rate at which the elevator rises ($5$ feet per second)
• Vertical - intercept: Represents the initial position of the elevator ($40$ feet below ground level)

Question 10:

• Function: $s(b)=\frac{5}{4}b$
• Slope: Represents the thickness of each book ($\frac{5}{4}$ inches per book)
• Vertical - intercept: Represents the height of the stack when there are no books ($0$ inches)