Use the dimensions of the container found in Part A.
C
c
What is the ratio of the three dimensions when x=3? Compare it to the ratio found in Part B.
A
a
Length: 3x+4 Width: 2x+1
B
b
2:5:10
C
c
No.
Practice makes perfect
a
We know that the volume of the shipping container can be represented by the polynomial 6x^3+11x^2+4x, where the height is x. To find the length and the width of the container, we will write this polynomial in the factored form. Let's start by factoring out x.
Next, we will factor the quadratic trinomial 6x^2+11x+4. It is in the form ax^2+bx+c, where |a| ≠1 and there are no common factors. To factor this expression, we will rewrite the middle term bx as two terms. The coefficients of these two terms will be factors of ac whose sum must be b.
x( 6x^2+11x+4)
We have that a= 6, b=11, and c=4. There are now three steps we need to follow in order to rewrite the above expression.
Find a c. Since we have that a= 6 and c=4, the value of ac is 6* 4=24.
Find factors of a c. Since ac=24, which is positive, we need factors of a c to have the same sign — both positive or both negative — in order for the product to be positive. Since b=11, which is also positive, those factors will need to be positive so that their sum is positive.
We have that the length of the container is (3x+4) and the width of the container is (2x+1).
b
We want to find the ratio of the three dimensions when x=2. We know that x is the height of the container. In Part A we have found that the width of the container is 2x+1 and that the length of the container is 3x+4.
Height:Width:Length
⇕
x:2x+1:3x+4
Finally, let's substitute x=2 into the above expression and simplify.
The ratio of the three dimensions when x=2 is 2:5:10.
c
In Part B we found the ratio of the three dimensions of the container when x=2. We want to determine whether it is the same for all values of x. To do so, let's substitute another x-value into the formula from Part B.
Height:Width:Length
⇕
x:2x+1:3x+4
We will substitute x=3 into the above expression and simplify.
The ratio of the three dimensions when x=3 is 3:7:13. It is different from the ratio found in Part B, so the ratio of the three dimensions will not be the same for all values of x.
