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A runner's speed is constant. In 2 hours, they run 48 km. In 7 hours, they run 168 km. How many hours will it take them to run 384 km? A garden party costs 2,000,000 UGX to rent plus 20,000 UGX per pe

Mathematics

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Answer

b

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

Step-by-step solution

8 steps

Here are the solutions to the math problems:

Part a· 2 steps

After how many hours will it take Joshua to run the 384 km?

  1. Calculate Joshua's speed.

    From the first record:

    Speed1=DistanceTime=48 km2 hours=24 km/hour\text{Speed}_1 = \frac{\text{Distance}}{\text{Time}} = \frac{48 \text{ km}}{2 \text{ hours}} = 24 \text{ km/hour}

    From the second record:

    Speed2=DistanceTime=168 km7 hours=24 km/hour\text{Speed}_2 = \frac{\text{Distance}}{\text{Time}} = \frac{168 \text{ km}}{7 \text{ hours}} = 24 \text{ km/hour}

    Joshua's consistent speed is 24 km/hour24 \text{ km/hour}.

  2. Calculate the time to run 384 km.

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

    It will take Joshua 16 hours to run 384 km.
    The information about "from 2023 to 2026" is irrelevant for this calculation.

Part b

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

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

  1. Identify fixed and variable costs.

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

  2. Formulate the equation.

    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

Find the total cost for the 20 people attend.

  1. Substitute P=20P=20 into the cost equation from part (b).

    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 d· 2 steps

Find the greatest number of the people the committee is able to invite.

  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.

    Subtract 2,000,000 from both sides:

    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

    Divide by 20,000:

    P≤3,000,00020,000P \le \frac{3,000,000}{20,000} P≤150P \le 150

    The greatest number of people the committee can invite is 150 people.

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