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Solve the following problems: a) Joshua runs 48 km in 2 hours and 168 km in 7 hours. How long will it take him to run 384 km? b) Write an equation for the total cost C of an evening party if the garde

Mathematics

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4 parts
a

16 hours\text{16 hours}

b

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

c

UGX 2,400,000\text{UGX 2,400,000}

d

150 people150 \text{ people}

Step-by-step solution

8 steps

Here are the solutions to the problems:

Part a· 2 steps

  1. Calculate Joshua's running rate.

    Joshua runs 48 km in 2 hours, so his rate is:

    Rate=48 km2 hours=24 km/hour\text{Rate} = \frac{48 \text{ km}}{2 \text{ hours}} = 24 \text{ km/hour}

    He also runs 168 km in 7 hours, which gives the same rate:

    Rate=168 km7 hours=24 km/hour\text{Rate} = \frac{168 \text{ km}}{7 \text{ hours}} = 24 \text{ km/hour}

  2. Calculate the time taken to run 384 km.

    Using the formula Time=DistanceRate\text{Time} = \frac{\text{Distance}}{\text{Rate}}:

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

    It will take Joshua 16 hours to run 384 km.

    16 hours\boxed{\text{16 hours}}

Part b

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

  1. Identify the fixed cost and variable cost.

    The cost of renting a party garden (fixed cost) is UGX 2,000,000.
    The cost of food per person (variable cost) is UGX 20,000.

  2. Write the equation.

    Let CC be the total cost and PP be the number of people.

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

    The equation connecting the total cost of the evening party with the number of people is C=2,000,000+20,000P\boxed{C = 2,000,000 + 20,000P}.

Part c

UGX 2,400,000\text{UGX 2,400,000}

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

    For 20 people, P=20P = 20. Using the 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=2,400,000 UGXC = 2,400,000 \text{ UGX}

    The total cost for 20 people is UGX 2,400,000\boxed{\text{UGX 2,400,000}}.

Part d

150 people150 \text{ people}

  1. Set up an inequality using the maximum budget.

    The maximum budget is UGX 5,000,000. So, the total cost CC must be less than or equal to this amount:

    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 is able to invite is 150 people\boxed{150 \text{ people}}.

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