Answer
(a) Dimensions = $5$ cm $\times$ $5$ cm
(b) $7500$ cm$^3$
Work Step by Step
(a) Based on the illustration, we know the following:
Length = $(60-2x)$ cm
Width = $(40-2x)$ cm
Area of the bottom of the container = $1500 cm^2$
We know that the formula for the area of a rectangle is Length $\times$ Width.
We can use this formula to form a quadratic equation!
Area = Length $\times$ Width
$1500 = (40-2x)(60-2x)$
$1500=4x^2-200x+2400$
(Subtract 1500 from both sides to make the equation=0)
$4x^2-200x+900=0$
We can then solve this equation by factoring!
$4x^2-180x-20x+900=0$
$4x(x-45)-20(x-45)=0$
$(4x-20)(x-45)=0$
We then need to make each bracket = 0 to find the solutions for $x$.
$(4x-20)=0$ or $(x-45)=0$
$x=5$ or $x=45$
We now have 2 solutions for $x$, but we can rule out 45 as if we substitute 45 into one of the dimensions, e.g: $60-2(45)$, we would get a length below 0 which is impossible. So, $x=5$
Dimensions = $5$cm $\times$ $5$ cm
(b) For the volume, we need to figure out the length, width, and height.
Length = $(60-2x)$
=$(60-2(5))$
=$50$ cm
Width = $(40-2x)$
=$(40-2(5))$
=$30$ cm
Height = $5$ cm (As this is the dimension of the squares cut out from each side)
So, we can find the volume using the formula Volume = $lwh$
V = $lwh$
V = $(50)(30)(5)$
V = $7500$ cm$^3$
