GEOMETRY
19. The area of a rectangle is represented by the function . What is the width of the rectangle if the length is represented by ?

Ask by Daniels Tran. in the United States

Mar 27,2025

Question

GEOMETRY 19. The area of a rectangle is represented by the function $$ f(x)=12 x^{2}+24 x-15 $$. What is the width of the rectangle if the length is represented by $$ g(x)=2 x+5 $$ ?

UpStudy AI · Accepted Answer

1. We are given the area of the rectangle as 
   $$
   f(x)=12x^{2}+24x-15
   $$
   and the length as 
   $$
   g(x)=2x+5.
   $$
   Since the area of a rectangle is the product of its length and width, if we let the width be $$ w(x) $$, we have
   $$
   (2x+5) \cdot w(x)=12x^{2}+24x-15.
   $$

2. To find the width $$ w(x) $$, divide the area by the length:
   $$
   w(x)=\frac{12x^{2}+24x-15}{2x+5}.
   $$

3. Perform polynomial long division:
   - Divide the leading term $$ 12x^{2} $$ by $$ 2x $$ to get $$ 6x $$.

- Multiply the divisor by $$ 6x $$:
     $$
     6x(2x+5)=12x^{2}+30x.
     $$

- Subtract this product from the area polynomial:
     $$
     (12x^{2}+24x-15) - (12x^{2}+30x) = -6x-15.
     $$

- Divide the new leading term $$ -6x $$ by $$ 2x $$ to get $$ -3 $$.

- Multiply the divisor by $$ -3 $$:
     $$
     -3(2x+5)=-6x-15.
     $$

- Subtract to find the remainder:
     $$
     (-6x-15)-(-6x-15)=0.
     $$

4. The quotient from the division is 
   $$
   w(x)=6x-3.
   $$

Thus, the width of the rectangle is 
$$
6x-3.
$$
The width of the rectangle is $$ 6x - 3 $$.

GEOMETRY
19. The area of a rectangle is represented by the function . What is the width of the rectangle if the length is represented by ?

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GEOMETRY 19. The area of a rectangle is represented by the function . What is the width of the rectangle if the length is represented by ?