The length of a rectangle is 2/3 mile. If the area is 1/2 square mile, what is the width?

Question

Aaliyah Anderson · Accepted Answer

Using first principles: #" area "=1/2 " square mile"#

Once you are used to these you should be doing some of the work in your head and write just a few lines.

If any doubt and the question lends itself to it draw a little sketch. This helps in understanding what to do. Does not need to all neat and fancy like mine. A very quick and rough sketch will do.

#"area " ="length "xx" width" color(magenta)( larr" Important to remember this")#
#color(white)("d")1/2color(white)("dd")= color(white)("d")2/3color(white)("dd.d")xx color(white)("dd")wcolor(white)("dd") larr " Given in the question"#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine the value of "w)#

#color(green)(1/2=2/3w)#

This is the same as:

#color(green)(1/2=2xx1/3xxw)#

Multiply both sides by #color(red)(1/2)#

#color(green)(ubrace(1/2color(red)(xx1/2))=ubrace(2color(red)(xx1/2))xx1/3xxw)#

#color(green)(color(white)("dd")1/4color(white)("dd")=color(white)("d..")ubrace(2/color(red)(2)color(white)("dd")xx1/3)xxw)#

#color(green)(color(white)("dd")1/4color(white)("dd")=color(white)("ddddd")1/3color(white)("ddd")xxw)#

Multiply both sides by #color(red)(3)#

#color(green)( color(red)(3)/4 = ubrace(color(red)(3)/3xx w)#

#color(green)(3/4 = color(white)("dd")w )#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine the area")#

#"area = length x width"#

#color(white)("d")a= 2/3xx3/4color(white)("dd") =color(white)("dd") (2xx3)/(3xx4) color(white)("dd")=color(white)("dd")6/12#

#a = (6-:6)/(12-:6)color(white)("dd") =color(white)("dd") 1/2 # square mile

Avery Allen · Answer

undefined To find the width of the rectangle, we can use the formula for the area of a rectangle, which is length multiplied by width. Given that the length of the rectangle is $ \frac{2}{3} $ mile and the area is $ \frac{1}{2} $ square mile, we can set up the equation:

$$ \text{Area} = \text{Length} \times \text{Width} $$

$$ \frac{1}{2} = \frac{2}{3} \times \text{Width} $$

To solve for the width, we can divide both sides of the equation by $ \frac{2}{3} $:

$$ \text{Width} = \frac{\frac{1}{2}}{\frac{2}{3}} $$

$$ \text{Width} = \frac{1}{2} \times \frac{3}{2} $$

$$ \text{Width} = \frac{3}{4} $$

So, the width of the rectangle is $ \frac{3}{4} $ mile.