A project manager uses the linear equation ax + by = c to model the allocation of resource… — SAT

A project manager uses the linear equation $ax + by = c$ to model the allocation of resources, where $x$ represents the number of hours spent on research, $y$ represents the number of hours spent on development, and $a, b,$ and $c$ are positive constants. Initially, the project manager allocates $40$ hours for research and $30$ hours for development. To optimize the project, the manager decides to increase the research hours by $25\%$ and decrease the development hours by $20\%$ such that the total resource value, $c$ , remains unchanged. Which of the following equations correctly expresses $a$ in terms of $b$ ?

A
$a = 1.67b$
$a = 0.6b$ Cevap
C
$a = 0.64b$
D
$a = 1.6b$

Cevap

The equation

a = 0.6b

correctly expresses

a

in terms of

b

The correct equation is

a = 0.6b

. The initial resource allocation is represented by

40a + 30b = c

. Increasing

40

25\%

gives

50

, and decreasing

30

20\%

gives

24

, which yields the new equation

50a + 24b = c

. Equating these two expressions since

c

is constant gives

40a + 30b = 50a + 24b

. Simplifying this equation results in

6b = 10a

, which gives

a = 0.6b

Adım Adım Çözüm

Write the linear equation representing the initial allocation of resources.

40a + 30b = c

The initial values are

40

hours for research (

x

) and

30

hours for development (

y

Calculate the new values of

x

and

y

after the percentage changes are applied.

x = 40 \times 1.25 = 50

and

y = 30 \times 0.80 = 24

Research hours increase by

25\%

and development hours decrease by

20\%

Write the new linear equation with the updated resource allocations.

50a + 24b = c

The total resource value

c

remains unchanged.

Equate the two expressions for

c

and solve for

a

in terms of

b

40a + 30b = 50a + 24b \implies 6b = 10a \implies a = 0.6b

Since both equations equal the same constant

c

, we set them equal to each other and isolate the variable

a

Anahtar Kavram

Expressing and manipulating linear relationships in two variables with constraints.

Alternatif Yöntem

Instead of keeping

c

general, we can choose a convenient value for

c

, such as

c = 120

. This gives two linear equations in terms of

a

and

b

40a + 30b = 120

and

50a + 24b = 120

. Solving this system for the ratio of

a

b

yields

a = 1.33

and

b = 2.22

, which confirms that

a/b = 0.6

, or

a = 0.6b

Tahmini Süre:3m 0s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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Alternatif Yöntem

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