Rule: W(x)=27x^3-12x Value: W(5)=3315 Interpretation: The volume of the pyramid on the diagram when x is 5 yards is 3315 cubic feet.
Practice makes perfect
We are given that V(x)=x^3-4x and W(x)=V(3x).
We are asked to find and interpret W(5).
Finding W(x)
Let's first find W(x) by replacing x with 3x in the definition of V.
V( x)= x^3-4 x âźą V( 3x)=( 3x)^3-4( 3x)After simplification, this gives the rule for W.
To interpret this value, let's start with the meaning of V(x).
Expression
Meaning
V(x)=x^3-4x
Volume (in cubic feet) of the pyramid on the diagram if x is measured in feet.
W(x)=27x^3-12x
Volume (in cubic feet) of the pyramid on the diagram if x is measured in yards.
We can hence interpret W(5)=3315 as the volume in cubic feet of the pyramid on the diagram when x is 5 yards.
Checking Our Answer
Checking our answer.
Let's use the expressions on the diagram to find the dimensions (in feet) of the pyramid when x is 5 yards.
Since some measurements are given in yards and some are in feet, we will need to use that there are 3 feet in a yard.
Description
Calculation
Length
First leg of the right-angled triangular base (2x-4). Four feet less than two times 5 yards.
2*5(3)-4 feet
26 feet
Second leg of the right-angled triangular base (3x+6). Six feet more than three times 5 yards.
3*5(3)+6 feet
51 feet
Height of the pyramid, x=5 yards.
5(3) feet
15 feet
Using these values, we can find the area of the base as half the product of the length of the legs.
A=1/2* 26* 51=663square feet
The volume of a pyramid is the third of the product of the area of the base and the height.
V=1/3* 663* 15=3315cubic feet
This result confirms the value we got for W(5).
