An automated pharmaceutical freeze-drying facility uses two types of sublimation condenser… — GMAT

An automated pharmaceutical freeze-drying facility uses two types of sublimation condensers, Unit Type A and Unit Type B, to process frozen liquid formulations into powder.

- Under standard mode, 4 units of Type A and 5 units of Type B operating simultaneously for 8 hours process a combined total of $5,120\text{ kg}$ of formulation.
- Under high-efficiency mode, the hourly processing rate of each Type A unit increases by $20\%$ , while the hourly processing rate of each Type B unit decreases by $15\%$ . When operating in high-efficiency mode, 3 units of Type A and 8 units of Type B running simultaneously for 5 hours process a combined total of $3,570\text{ kg}$ of formulation.

Select the hourly processing rate (in kg/hr) for a single Type A unit operating under high-efficiency mode, and the hourly processing rate (in kg/hr) for a single Type B unit operating under standard mode.

Hourly processing rate of 1 Type A unit under high-efficiency mode (kg/hr)102
Hourly processing rate of 1 Type B unit under standard mode (kg/hr)60

Cevap

High-efficiency rate of Type A = 102 kg/hr; Standard rate of Type B = 60 kg/hr.

Solving the system of linear equations yields standard rates of

85\text{ kg/hr}

for Type A and

60\text{ kg/hr}

for Type B. Applying the

20\%

increase to Type A gives

1.20 \times 85 = 102\text{ kg/hr}

for the high-efficiency Type A column, while the standard Type B column requires

60\text{ kg/hr}

Adım Adım Çözüm

Define variables for the standard hourly rates of Type A and Type B units.

Let

x

be the standard hourly processing rate of 1 Type A unit (in kg/hr), and let

y

be the standard hourly processing rate of 1 Type B unit (in kg/hr).

Establishing individual unit rates allows setting up linear equations based on total mass processed.

Formulate the first linear equation using standard mode data.

Total hourly rate for 4 Type A and 5 Type B units is

4x + 5y

. Operating for 8 hours gives

8(4x + 5y) = 5,120 \implies 4x + 5y = 640

Dividing total mass by hours yields the combined hourly processing capacity of the configuration.

Formulate the second linear equation using high-efficiency mode data.

High-efficiency Type A rate

= 1.20x

, and high-efficiency Type B rate

= 0.85y

.
For 3 Type A and 8 Type B units running 5 hours:

5(3(1.20x) + 8(0.85y)) = 3,570 \implies 5(3.6x + 6.8y) = 3,570 \implies 3.6x + 6.8y = 714

.
Multiply by 10 to clear decimals:

36x + 68y = 7,140 \implies 9x + 17y = 1,785

Applying percentage rate modifications gives the adjusted unit rates for high-efficiency mode.

Solve the system of simultaneous linear equations for

x

and

y

From Equation 1 (

4x + 5y = 640

), express

x = \frac{640 - 5y}{4}

.
Substitute into Equation 2 (

9x + 17y = 1,785

9\left(\frac{640 - 5y}{4}\right) + 17y = 1,785

5,760 - 45y + 68y = 7,140

23y = 1,380 \implies y = 60

.
Substituting

y = 60

back into

4x + 5(60) = 640 \implies 4x = 340 \implies x = 85

Using substitution resolves the two-variable system to find the standard rates.

Calculate the target quantities requested in the prompt.

Standard rate of Type B

= y = 60\text{ kg/hr}

.
High-efficiency rate of Type A

= 1.20x = 1.20 \times 85 = 102\text{ kg/hr}

The question specifically asks for Type A's high-efficiency rate and Type B's standard rate.

Anahtar Kavram

Formulating and solving simultaneous multi-variable linear systems derived from work rate scenarios with non-standard modifiers.

Alternatif Yöntem

Clear the hourly rates immediately: Equation 1 gives combined rate

4x + 5y = 640

. Equation 2 gives

3.6x + 6.8y = 714

. Multiply Equation 1 by

0.9

to align

x

-coefficients (

3.6x + 4.5y = 576

), then subtract from Equation 2:

2.3y = 138 \implies y = 60

Tahmini Süre:2m 30s

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