unit review a: foundations for functions
problem 11 at the county beach, the lifeguard policy states that one lifeguard is required for up to 50
swimmers, with an additional lifeguard required for each additional 25 swimmers. the maximum allowed
swimmers, s, in the water given l, the number of lifeguards on duty is represented by the equation
s = 50 + 25(l - 1). the beach schedules up to 8 lifeguards on busy days.
a. what function family represents this situation? explain how you know.

b. does discrete or continuous domain and
range best represent this situation? why?
c. determine the domain and range of this
situation.

d. find the inverse of the equation
representing this situation.
e. use the inverse to calculate the number of
lifeguards required to accommodate 400
swimmers.

f. is this within the domain of the situation? what options do the beach managers have to address this
situation?

problem 12 will is making the 50 minute drive home from an away game. the distance remaining is
represented by the equation d = 54 - 1.08t, where t is the time in minutes.

a. does discrete or continuous domain and
range best represent this situation? why?
b. determine the domain and range of this
situation.

c. find the inverse of the equation
representing this situation.
d. use the inverse to calculate how long it
will take will to reach the halfway point (23
miles)

math beach solutions llc 2018

Question

unit review a: foundations for functions
problem 11 at the county beach, the lifeguard policy states that one lifeguard is required for up to 50
swimmers, with an additional lifeguard required for each additional 25 swimmers. the maximum allowed
swimmers, s, in the water given l, the number of lifeguards on duty is represented by the equation
s = 50 + 25(l - 1). the beach schedules up to 8 lifeguards on busy days.
a. what function family represents this situation? explain how you know.

b. does discrete or continuous domain and
   range best represent this situation? why?
c. determine the domain and range of this
   situation.

d. find the inverse of the equation
   representing this situation.
e. use the inverse to calculate the number of
   lifeguards required to accommodate 400
   swimmers.

f. is this within the domain of the situation? what options do the beach managers have to address this
   situation?

problem 12 will is making the 50 minute drive home from an away game. the distance remaining is
represented by the equation d = 54 - 1.08t, where t is the time in minutes.

a. does discrete or continuous domain and
   range best represent this situation? why?
b. determine the domain and range of this
   situation.

c. find the inverse of the equation
   representing this situation.
d. use the inverse to calculate how long it
   will take will to reach the halfway point (23
   miles)

math beach solutions llc 2018

Accepted Answer

### Problem 11
#### Part A
# Explanation:
## Step1: Analyze the equation form
The given equation is $ S = 50 + 25(L - 1) $. Simplify it: $ S=50 + 25L-25=25L + 25 $. This is in the form of a linear equation $ y=mx + b $ (where $ m = 25 $, $ b = 25 $, $ y = S $, $ x = L $).
## Step2: Identify function family
Linear functions have the form $ y=mx + b $ with a constant rate of change (here, the rate of change of swimmers with respect to lifeguards is 25, constant). So the function family is linear.
# Answer:
The function family is linear. The equation simplifies to $ S = 25L + 25 $, which is in the form $ y=mx + b $ (linear function) with a constant slope (rate of change).

#### Part B
# Explanation:
## Step1: Define discrete/continuous
Discrete data consists of distinct, separate values (like counts of lifeguards/swimmers, which must be whole numbers). Continuous data can take any value in an interval.
## Step2: Apply to situation
The number of lifeguards $ L $ is a count (1, 2, ..., 8) and swimmers $ S $ is also a count (based on 25 - swimmer increments from 50). These are distinct whole numbers, so discrete domain (for $ L $) and range (for $ S $) best represent the situation.
# Answer:
Discrete. Because the number of lifeguards ($ L $) and swimmers ($ S $) are counts of whole numbers (you can't have a fraction of a lifeguard or swimmer), so their values are distinct and separate, not continuous.

#### Part C
# Explanation:
## Step1: Find domain (Lifeguards, L)
The beach schedules up to 8 lifeguards, and $ L $ must be at least 1 (since one lifeguard is required for up to 50 swimmers). So $ L $ is in $ \{1, 2, 3, 4, 5, 6, 7, 8\} $ (or $ 1\leq L\leq8 $, $ L\in\mathbb{Z} $).
## Step2: Find range (Swimmers, S)
Use the equation $ S = 50 + 25(L - 1) $. For $ L = 1 $: $ S = 50 $. For $ L = 8 $: $ S = 50 + 25(7)=50 + 175 = 225 $. The values of $ S $ are $ 50, 75, 100, ..., 225 $ (each time $ L $ increases by 1, $ S $ increases by 25). So the range is $ \{50, 75, 100, 125, 150, 175, 200, 225\} $ (or $ 50\leq S\leq225 $, $ S = 50 + 25k $, $ k = 0,1,...,7 $).
# Answer:
Domain: $ L\in\{1, 2, 3, 4, 5, 6, 7, 8\} $ (or $ 1\leq L\leq8 $, $ L $ is an integer).
Range: $ S\in\{50, 75, 100, 125, 150, 175, 200, 225\} $ (or $ 50\leq S\leq225 $, $ S = 50 + 25(L - 1) $ for $ L $ in the domain).

#### Part D
# Explanation:
## Step1: Start with $ S = 50 + 25(L - 1) $
Solve for $ L $ in terms of $ S $. First, expand: $ S = 50 + 25L - 25 \Rightarrow S = 25L + 25 $.
## Step2: Isolate L
Subtract 25: $ S - 25 = 25L $. Divide by 25: $ L=\frac{S - 25}{25}=\frac{S}{25}-1 $.
# Answer:
The inverse function is $ L=\frac{S}{25}-1 $ (or $ L=\frac{S - 25}{25} $).

#### Part E
# Explanation:
## Step1: Use inverse function
We need to find $ L $ when $ S = 400 $. Use $ L=\frac{S}{25}-1 $.
## Step2: Substitute S = 400
$ L=\frac{400}{25}-1 = 16 - 1 = 15 $. Wait, but the domain of $ L $ is up to 8. Wait, no—wait, the original problem: the beach schedules up to 8 lifeguards. Wait, maybe I made a mistake? Wait, no, the inverse is for calculating $ L $ given $ S $, but we must check if $ S = 400 $ is in the range? Wait, no, the question is to use the inverse to calculate $ L $ for $ S = 400 $, regardless of the domain (but in reality, the beach can't have 15 lifeguards as per their schedule, but mathematically, using the inverse: $ L=\frac{400 - 25}{25}=\frac{375}{25}=15 $. Wait, no, earlier expansion: $ S = 25L + 25 $, so $ L=(S - 25)/25 $. For $ S = 400 $, $ L=(400 - 25)/25 = 375/25 = 15 $. But the beach's schedule is up to 8 lifeguards, so this $ S = 400 $ is outside the original range (max $ S $ with 8 lifeguards is 225). But the question just asks to use the inverse to calculate, so mathematically:
# Answer:
Using the inverse $ L=\frac{S - 25}{25} $, substitute $ S = 400 $: $ L=\frac{400 - 25}{25}=\frac{375}{25}=15 $. So the number of lifeguards required (mathematically) is 15.

#### Part F
# Explanation:
## Step1: Check domain of L
The original domain of $ L $ is $ 1\leq L\leq8 $ (integer). The calculated $ L = 15 $ is greater than 8, so it is not within the domain of the situation (beach only schedules up to 8 lifeguards).
## Step2: Beach managers' options
To address this, they could: 1) Increase the maximum number of lifeguards allowed (if possible, e.g., hire more, adjust policy). 2) Limit the number of swimmers to stay within the 8 - lifeguard capacity (e.g., close off parts of the beach, implement a swimmer limit).
# Answer:
No, $ L = 15 $ is not within the domain ($ 1\leq L\leq8 $). Options for managers: Increase the maximum number of lifeguards (if feasible) or limit the number of swimmers to stay within the 8 - lifeguard capacity.

### Problem 12
#### Part A
# Explanation:
## Step1: Define discrete/continuous
Time $ t $ (in minutes) and distance $ d $ (in miles) can take any value in an interval (e.g., $ t = 2.5 $ minutes, $ d = 54 - 1.08(2.5) $ miles). They are not restricted to whole numbers; time and distance can be fractional.
## Step2: Apply to situation
Since $ t $ (time) and $ d $ (distance) can take any real - number value in their respective intervals (e.g., 0.5 minutes, 23.5 miles), the domain (for $ t $) and range (for $ d $) are continuous.
# Answer:
Continuous. Because time ($ t $) and distance ($ d $) can take any value in their intervals (e.g., fractional minutes or miles), not just whole numbers, so they are continuous.

#### Part B
# Explanation:
## Step1: Find domain (Time, t)
The drive is 50 minutes, so $ t $ starts at 0 (when he starts driving) and ends at 50 (when he arrives). So $ 0\leq t\leq50 $, $ t\in\mathbb{R} $.
## Step2: Find range (Distance, d)
Use the equation $ d = 54 - 1.08t $. When $ t = 0 $: $ d = 54 $ (starting distance). When $ t = 50 $: $ d = 54 - 1.08(50)=54 - 54 = 0 $. The function $ d(t) $ is linear with a negative slope, so as $ t $ increases from 0 to 50, $ d $ decreases from 54 to 0. So the range is $ 0\leq d\leq54 $, $ d\in\mathbb{R} $.
# Answer:
Domain: $ 0\leq t\leq50 $ (where $ t $ is time in minutes, $ t\in\mathbb{R} $).
Range: $ 0\leq d\leq54 $ (where $ d $ is distance in miles, $ d\in\mathbb{R} $).

#### Part C
# Explanation:
## Step1: Start with $ d = 54 - 1.08t $
Solve for $ t $ in terms of $ d $. First, subtract 54 from both sides: $ d - 54=-1.08t $.
## Step2: Isolate t
Divide both sides by - 1.08: $ t=\frac{54 - d}{1.08}=\frac{54}{1.08}-\frac{d}{1.08}=50-\frac{d}{1.08} $ (or $ t=\frac{54 - d}{1.08} $).
# Answer:
The inverse function is $ t=\frac{54 - d}{1.08} $ (or $ t = 50-\frac{d}{1.08} $).

#### Part D
# Explanation:
## Step1: Half - way point distance
The halfway point is 23 miles (given). We use the inverse function $ t=\frac{54 - d}{1.08} $.
## Step2: Substitute d = 23
$ t=\frac{54 - 23}{1.08}=\frac{31}{1.08}\approx28.70 $ minutes (or exact fraction: $ \frac{31}{1.08}=\frac{3100}{108}=\frac{775}{27}\approx28.70 $).
# Answer:
Using the inverse $ t=\frac{54 - d}{1.08} $, substitute $ d = 23 $: $ t=\frac{54 - 23}{1.08}=\frac{31}{1.08}\approx28.70 $ minutes (or $ \frac{775}{27}\approx28.70 $ minutes).

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QUESTION IMAGE

Question

Explanation:

Problem 11

Part A

Step1: Analyze the equation form

Step2: Identify function family

Step1: Define discrete/continuous

Step2: Apply to situation

Step1: Find domain (Lifeguards, L)

Step2: Find range (Swimmers, S)

Answer:

Part B