The operating cost C (in Naira per hour) of a speed boat is partly constant and partly var… — JAMB UTME

The operating cost $C$ (in Naira per hour) of a speed boat is partly constant and partly varies directly as the square of its speed $v$ (in km/h). At a speed of $20\text{ km/h}$ , the operating cost is $₦18,000$ per hour, and at a speed of $30\text{ km/h}$ , the operating cost is $₦33,000$ per hour. What is the operating cost per hour (in Naira) when the boat travels at a speed of $40\text{ km/h}$ ?

Cevap: 54000 Naira

Cevap

The operating cost per hour when the speed boat travels at

40\text{ km/h}

54,000

Naira.

By writing the partial variation relationship as

C = k_1 + k_2 v^2

and substituting the given pairs

(20, 18000)

and

(30, 33000)

, we set up simultaneous equations

k_1 + 400k_2 = 18000

and

k_1 + 900k_2 = 33000

. Solving these yields

k_2 = 30

and

k_1 = 6000

. Evaluating the model at

v = 40\text{ km/h}

gives

C = 6000 + 30(40^2) = 54,000

Naira per hour.

Adım Adım Çözüm

Set up the general formula for partial variation

C = k_1 + k_2 v^2

The total cost consists of a fixed constant component

k_1

and a variable component

k_2 v^2

that varies directly with speed squared.

Form simultaneous linear equations from the given conditions

18,000 = k_1 + 400 k_2

and

33,000 = k_1 + 900 k_2

Substituting

v = 20, C = 18,000

and

v = 30, C = 33,000

creates a solvable system of equations in

k_1

and

k_2

Solve for the variation constants

k_1

and

k_2

k_2 = 30

and

k_1 = 6,000

Subtracting the first equation from the second eliminates

k_1

, giving

500 k_2 = 15,000 \implies k_2 = 30

. Substituting

k_2 = 30

into the first equation yields

k_1 = 6,000

Calculate the operating cost at

v = 40\text{ km/h}

C = 54,000

Evaluating

C = 6,000 + 30(40)^2 = 6,000 + 30(1,600) = 6,000 + 48,000 = 54,000

Anahtar Kavram

Partial Variation with Simultaneous Equations

Soru

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