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Joshua runs at a constant speed. He ran 48 km in 2 hours and 168 km in 7 hours. How long will it take him to run 384 km? In part b, calculate the total cost of an evening party for P people if the gar

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2 parts
b

C=2,000,000+20,000PC = 2,000,000 + 20,000P

d
150

Step-by-step solution

9 steps

Part a· 2 steps

  1. Calculate Joshua's running speed.

    Joshua's speed is consistent.

    Speed=DistanceTime\text{Speed} = \frac{\text{Distance}}{\text{Time}}

    Using the first record:

    Speed1=48 km2 hours=24 km/h\text{Speed}_1 = \frac{48 \text{ km}}{2 \text{ hours}} = 24 \text{ km/h}

    Using the second record:

    Speed2=168 km7 hours=24 km/h\text{Speed}_2 = \frac{168 \text{ km}}{7 \text{ hours}} = 24 \text{ km/h}

    Joshua's constant speed is 24 km/h24 \text{ km/h}.

  2. Calculate the time it will take Joshua to run 384 km.

    Time=DistanceSpeed\text{Time} = \frac{\text{Distance}}{\text{Speed}} Time=384 km24 km/h\text{Time} = \frac{384 \text{ km}}{24 \text{ km/h}} Time=16 hours\text{Time} = 16 \text{ hours}

    It will take Joshua 16 hours to run 384 km.
    The information about "from 2023 to 2026" is extraneous for this calculation, as the question asks for the time to run the distance, not the year he completes it.

Part b

C=2,000,000+20,000PC = 2,000,000 + 20,000P

  1. Identify the fixed cost and the variable cost per person.

    Fixed cost (renting garden) = UGX 2,000,000
    Cost per person (food) = UGX 20,000
    Let CC be the total cost and PP be the number of people.

  2. Write the equation connecting the total cost of the evening party with the number of people.

    C=2,000,000+20,000PC = 2,000,000 + 20,000P

    The equation is C=2,000,000+20,000P\boxed{C = 2,000,000 + 20,000P}.

Part c· 2 steps

  1. Substitute the number of people into the cost equation.

    For 20 people, P=20P = 20.

    C=2,000,000+20,000(20)C = 2,000,000 + 20,000(20)

  2. Calculate the total cost.

    C=2,000,000+400,000C = 2,000,000 + 400,000 C=2,400,000C = 2,400,000

    The total cost for 20 people is UGX 2,400,000.

Part d150

  1. Set up an inequality using the maximum budget.

    The maximum budget is UGX 5,000,000.

    2,000,000+20,000P≤5,000,0002,000,000 + 20,000P \le 5,000,000

  2. Solve the inequality for PP.

    20,000P≤5,000,000−2,000,00020,000P \le 5,000,000 - 2,000,000 20,000P≤3,000,00020,000P \le 3,000,000 P≤3,000,00020,000P \le \frac{3,000,000}{20,000} P≤150P \le 150

  3. Determine the greatest number of people.

    Since the number of people must be a whole number, the greatest number of people the committee is able to invite is 150.
    The greatest number of people is 150\boxed{150}.

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