Let's break down the given problems and present detailed step-by-step solutions.
### Problem 9:
The airplane is flying at an initial height of 35,000 feet above sea level. It starts to descend at a rate of 2,500 feet per minute. We need to find an expression that describes the height of the airplane after any given number of minutes, denoted by [tex]\( m \)[/tex].
1. Identify the initial height:
[tex]\[ \text{Initial height} = 35,000 \text{ feet} \][/tex]
2. Identify the rate of descent:
[tex]\[ \text{Rate of descent} = 2,500 \text{ feet per minute} \][/tex]
3. Determine the height change over time:
- As the airplane descends, it loses height.
- After [tex]\(m\)[/tex] minutes, the airplane will have descended [tex]\( 2,500 \times m \)[/tex] feet.
4. Formulate the expression for the height after [tex]\( m \)[/tex] minutes:
[tex]\[ \text{Height after } m \text{ minutes} = \text{Initial height} - (\text{Rate of descent} \times m) \][/tex]
[tex]\[ \text{Height after } m \text{ minutes} = 35,000 - 2,500m \][/tex]
Thus, the correct expression is:
[tex]\[ 35,000 - 2,500m \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{C. \; 35,000 - 2,500m} \][/tex]
### Problem 10:
We need to create a storyline (word problem) using the given algebraic expressions:
- Part A: [tex]\(\frac{1,000}{r}\)[/tex]
- Part B: [tex]\(75 - 3m\)[/tex]
- Part C: [tex]\(30 + 2d\)[/tex]
Let's construct a cohesive storyline involving these expressions:
Storyline:
John runs a transportation company that deals with shipping packages.
- Part A: John charges a rate for each package he ships based on the speed of delivery, represented by [tex]\( r \)[/tex]. The cost to ship a package with priority delivery is given by [tex]\(\frac{1,000}{r}\)[/tex] dollars, where [tex]\( r \)[/tex] is the speed in miles per hour. Faster deliveries are more expensive.
- Part B: John has a driving route that covers multiple stops each day. He starts his daily route with 75 packages. At each stop, he delivers [tex]\( 3 \)[/tex] packages. If [tex]\( m \)[/tex] is the number of stops he makes, the number of packages remaining to be delivered after [tex]\( m \)[/tex] stops is given by [tex]\( 75 - 3m \)[/tex].
- Part C: John often needs to hire temporary workers for sorting packages at different locations. The total number of hours he needs for sorting packages, depending on the number of days [tex]\( d \)[/tex] the workers are hired, is represented by [tex]\( 30 + 2d \)[/tex]. This means he starts with a base need of 30 hours and hires for 2 additional hours per day.
This storyline tying together the expressions provides a realistic scenario that involves transportation logistics, package delivery, and temporary manpower allocation in John's business operations.
