Question

Suppose you want to build a small walkway around your garden so that there is someplace to walk around and work on your vegetables. The garden measures 20 feet by 30 feet and you want to increase the area of the garden to 1,000 ft2. What is the width of the walkway?
Approximately 3.3 feet
Approximately 3.5 feet
Approximately 3.7 feet
Approximately 3.9 feet

Answer 1

Answer:

Approximately 3.5 feet - Option B

Step-by-step explanation:

Imagine that you have this walkway around the garden, with dimensions 30 by 20 feet. This walkway has a difference of x feet between it's length, and say the dimension 30 feet. In fact it has a difference of x along both dimensions - on either ends. Therefore, the increases length and width should be 30 + 2x, and 20 + 2x, which is with respect to an increases area of 1,000 square feet.

( 30 + 2x ) $*$ ( 20 + 2x ) = 1000 - Expand "( 30 + 2x ) $*$ ( 20 + 2x )"

600 + 100x + 4$x^2$ = 1000 - Subtract 1000 on either side, making on side = 0

4$x^2$ + 100x - 400 = 0 - Take the "quadratic equation formula"  

( Quadratic Equation is as follows ) - $x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$,

$x=\frac{-100+\sqrt{100^2-4\cdot \:4\left(-400\right)}}{2\cdot \:4}:\quad \frac{5\left(\sqrt{41}-5\right)}{2}$,

$x=\frac{-100-\sqrt{100^2-4\cdot \:4\left(-400\right)}}{2\cdot \:4}:\quad -\frac{5\left(5+\sqrt{41}\right)}{2}$

There can't be a negative width of the walkway, hence our solution should be ( in exact terms ) $\frac{5\left(\sqrt{41}-5\right)}{2}$. The approximated value however is 3.5081...or approximately 3.5 feet.