Question

final exam - requires respondus lockdown browser + webcam
started: dec 10 at 8:58pm
quiz instructions
question 29
1 pts
solve the problem.
the length of a rectangular flower bed is 7 ft. less than 3 times its width. the area of the bed is 20 ft². find the width of the flower bed. do not enter units - just the number.

Explanation:

Step1: Define variables

Let the width of the flower bed be \( w \) feet. Then the length \( l \) is \( 3w - 7 \) feet (since length is 7 ft less than 3 times width).

Step2: Use area formula

The area \( A \) of a rectangle is \( A = l \times w \). We know \( A = 20 \), so substitute \( l \) and \( A \):

$$ (3w - 7) \times w = 20 $$

Expand to get a quadratic equation:

$$ 3w^2 - 7w - 20 = 0 $$

Step3: Solve quadratic equation

Factor the quadratic (or use quadratic formula). Let's factor:
We need two numbers that multiply to \( 3 \times (-20) = -60 \) and add to \( -7 \). The numbers are \( -12 \) and \( 5 \).
Rewrite the middle term:

$$ 3w^2 - 12w + 5w - 20 = 0 $$

Group and factor:

$$ 3w(w - 4) + 5(w - 4) = 0 \implies (3w + 5)(w - 4) = 0 $$

Set each factor to zero:
\( 3w + 5 = 0 \) gives \( w = -\frac{5}{3} \) (discard, width can't be negative), \( w - 4 = 0 \) gives \( w = 4 \).

Answer:

4