A logistics company uses two delivery vehicles, Vehicle P and Vehicle Q, to transport carg… — GRE General Test

A logistics company uses two delivery vehicles, Vehicle P and Vehicle Q, to transport cargo between two warehouses that are $240$ miles apart. Vehicle P travels at a constant average speed of $r$ miles per hour ( $r > 0$ ). Vehicle Q travels at a constant average speed that is $20$ miles per hour faster than that of Vehicle P. In addition to driving time, Vehicle P requires a flat setup time of $1$ hour before departure, while Vehicle Q requires a flat setup time of $2$ hours before departure.

Let $T_P(r)$ and $T_Q(r)$ represent the total elapsed time in hours (including setup time) required for Vehicle P and Vehicle Q to complete the trip, respectively.

Which of the following statements are true? Select all such statements.

The total elapsed time for Vehicle P, in hours, as a function of its speed $r$ , is given by $T_P(r) = \frac{240 + r}{r}$ .Cevap
Vehicle P and Vehicle Q take the exact same total elapsed time to complete the trip when $r = 60$ miles per hour.Cevap
C
For all positive speeds $r > 0$ , Vehicle Q always requires strictly less total elapsed time than Vehicle P to complete the trip.
D
If $r = 30$ miles per hour, the travel time (excluding setup time) for Vehicle Q is half the travel time (excluding setup time) for Vehicle P.
E
Doubling the speed of Vehicle P from $r = 40$ miles per hour to $r = 80$ miles per hour reduces its total elapsed time $T_P(r)$ by $50\%$ .

Cevap

The correct statements are the algebraic expression for Vehicle P's total time as a function of speed and the equal total elapsed time condition at a speed of 60 miles per hour.

The expression for Vehicle P's total time accurately combines

1

hour of setup time with

\frac{240}{r}

driving hours to get

\frac{240+r}{r}

. Setting

T_P(r) = T_Q(r)

yields the quadratic equation

r^2 + 20r - 4800 = 0

, which correctly solves to

r = 60

mph, at which point both vehicles require exactly

5

hours total.

Adım Adım Çözüm

Model total elapsed time functions

T_P(r)

and

T_Q(r)

using setup time plus travel time.

T_P(r) = 1 + \frac{240}{r} = \frac{240 + r}{r}

and

T_Q(r) = 2 + \frac{240}{r + 20}

Total elapsed time is the sum of fixed pre-departure setup overhead and variable driving time.

Determine the speed

r

where total elapsed times are equal by setting

T_P(r) = T_Q(r)

1 + \frac{240}{r} = 2 + \frac{240}{r + 20} \implies \frac{240}{r} - \frac{240}{r + 20} = 1 \implies 240 \cdot 20 = r(r + 20) \implies r^2 + 20r - 4800 = 0

. Factoring gives

(r - 60)(r + 80) = 0

, so

r = 60

mph.

Solving the rational equation identifies the exact breakeven speed where higher travel efficiency balances extra setup time.

Evaluate the remaining candidate assertions against the derived model.

For

r > 60

T_Q(r) > T_P(r)

, invalidating the claim that Vehicle Q is always faster. At

r = 40

T_P(40) = 7

hours and

T_P(80) = 4

hours, giving a decrease of

\frac{3}{7} \approx 42.86\%

, invalidating the

50\%

reduction claim.

Fixed setup costs distort simple constant-proportion and percentage changes in total time.

Anahtar Kavram

Algebraic Modeling of Combined Time with Fixed Overhead and Variable Rates

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Cevap

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