Answer:
Mix fraction: 3 1/27
Improper fraction: 82/27
Decimal approximation: 3.037
Step-by-step explanation:
What value is close to 28 that is a perfect cube...27 which equals 3^3.
So let's find the tangent line to the curve y=cubert(x) at x=27.
We will use this equation to approximate what happens at x=28.
First let's rewrite the radical in our equation;
y=x^(1/3)
Now differentiate
y'=(1/3) x^(1/3-1) by power rule
Simplify
y'=(1/3) x^(-2/3) or (1)/(3x^[2/3])
So the slope of our tangent line at x=27 is (1)/(3(27)^[2/3])=1/(3(3)^2)=1/(3×9)=1/27.
We will also need a point on this tangent line....We know we have the point at x=27 because that is what our tangent line to curve is being found at.
So at x=27, we have y=cubert(27)=3. We used our equation y=cubert(x) here.
So we want to find the equation of the line that contains point (27,3) and has slope 1/27.
Point-slope form is
y-y1=m(x-x1)
Plug in our values
y-3=(1/27)(x-27)
Add 3 on both sides
y=3+(1/27)(x-27)
We will use this linear equation to approximate cubert(28) by replacing x with 28.
y=3+(1/27)(28-27)
y=3+(1/27)(1)
y=3+1/27
You can write that as a mix fraction if you want.
This value is than 3 but super close to 3 since 1/27 is close to 0.
Mix fraction: 3 1/27
Improper fraction: 82/27
Decimal approximation: 3.037
Cubert of 28 when smashed into calculator as is gives approximately 3.0366 which is pretty close to our approximation.
