A landscape architect is designing a park layout. The total area of the park is represente… — ACT

A landscape architect is designing a park layout. The total area of the park is represented by the polynomial $A(t) = (2t - 3)(3t^2 + 4t - 1)$ square meters, while the area allocated for a parking lot is represented by $P(t) = 2t^2(3t - 5)$ square meters, where $t$ represents a scaling factor. The remaining green space in the park is modeled by the polynomial $g(t) = A(t) - P(t)$ . When $g(t)$ is written in the standard form $at^3 + bt^2 + ct + d$ , where $a$ , $b$ , $c$ , and $d$ are constants, what is the value of $b$ ?

Cevap: 9

Cevap

The value of

b

, the coefficient of the

t^2

term in the standard form of

g(t)

, is

9

Expanding the polynomials gives

A(t) = 6t^3 - t^2 - 14t + 3

and

P(t) = 6t^3 - 10t^2

. Subtracting

P(t)

from

A(t)

yields

g(t) = (6t^3 - t^2 - 14t + 3) - (6t^3 - 10t^2) = 9t^2 - 14t + 3

. In the standard form

at^3 + bt^2 + ct + d

, the coefficient

b

of the

t^2

term is

9

Adım Adım Çözüm

Expand the polynomial

A(t) = (2t - 3)(3t^2 + 4t - 1)

6t^3 - t^2 - 14t + 3

To represent the total area as a single polynomial in standard form before subtraction.

Expand the polynomial

P(t) = 2t^2(3t - 5)

6t^3 - 10t^2

To represent the parking lot area as a simplified polynomial in standard form.

Subtract

P(t)

from

A(t)

to find the green space polynomial

g(t)

9t^2 - 14t + 3

Subtracting

P(t)

from

A(t)

requires distributing the negative sign to both terms, yielding

-t^2 - (-10t^2) = 9t^2

Identify the coefficient

b

of the

t^2

term in

at^3 + bt^2 + ct + d

9

Comparing

g(t) = 9t^2 - 14t + 3

to the standard cubic form reveals that

a = 0

and

b = 9

Anahtar Kavram

Polynomial operations including expansion of products and subtraction with negative sign distribution

Soru

Cevap

Adım Adım Çözüm

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