The total cost C per trip of operating a high-speed passenger ferry consists of a fixed ad… — JAMB UTME

The total cost $C$ per trip of operating a high-speed passenger ferry consists of a fixed administrative overhead cost and an operational cost that varies directly as the cube of its speed $v$ in knots. Given that the total cost per trip is $\$1,400$ when the ferry travels at $10\text{ knots}$ and $\$4,200$ when it travels at $20\text{ knots}$ , what is the total cost per trip when the ferry operates at a speed of $15\text{ knots}$ ?

$\\$ 2,350$Cevap
B
$\\$ 2,800$
C
$\\$ 3,300$
D
$\\$ 4,725$

Cevap

\\

2,350$

The partial variation formula is

C = k_1 + k_2 v^3

. Substituting the given points yields

k_1 + 1000 k_2 = 1400

and

k_1 + 8000 k_2 = 4200

. Subtracting these equations gives

7000 k_2 = 2800

, so

k_2 = 0.4

and

k_1 = 1000

. Evaluating at

v = 15

gives

C = 1000 + 0.4(3375) = 1000 + 1350 = \\

2,350$.

Adım Adım Çözüm

Set up the partial variation equation

C = k_1 + k_2 v^3

Partial variation combines a fixed constant

k_1

with a variable term

k_2 v^3

Form simultaneous linear equations using given conditions

Equation (1):

k_1 + 1000 k_2 = 1400

; Equation (2):

k_1 + 8000 k_2 = 4200

Substitute

v = 10

C = 1400

and

v = 20

C = 4200

into the variation model.

Solve for the constants

k_1

and

k_2

k_2 = 0.4

and

k_1 = 1000

Subtract Equation (1) from Equation (2):

7000 k_2 = 2800 \implies k_2 = 0.4

. Substitute

k_2 = 0.4

into Equation (1) to get

k_1 = 1400 - 400 = 1000

Calculate total cost

C

for speed

v = 15\text{ knots}

C = 1000 + 0.4(15)^3 = 1000 + 0.4(3375) = 1000 + 1350 = 2350

Substitute

k_1 = 1000

k_2 = 0.4

, and

v = 15

back into the formula.

Anahtar Kavram

Partial Variation with non-linear powers solved via simultaneous linear equations

Tahmini Süre:2m 0s

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