Estimate the value of 3 to the power of 2.2 using linear interpolation.
3^2=9
3^3=27
linear extrapolation on an exponential function? Goodness.
3^2.2= 9+.2(27-9)/(1)= 9+.2*18=12.6
and the audience says the real value is ...11.2
The linear interpolation formula is given as:
y = y1 + ( x − x1 ) ∙ ( y2 − y1 ) / ( x2 − x1 )
2.2 is between 2 and 3
Get the starting values:
x1 = 2 , y1 = 3^2 = 9 , x2 = 3 , y2 = 3^3 = 27
y = y1 + ( x − x1 ) ∙ ( y2 − y1 ) / ( x2 − x1 )
y = 9 + ( x − 2 ) ∙ ( 27 − 9 ) / ( 3 − 2 )
y = 9 + ( x − 2 ) ∙ 18 / 1
y = 9 + ( x − 2 ) ∙ 18
y = 9 + 18 x − 36
y = - 27 + 18 x
y = 18 x - 27
for x = 2.2
y = 18 ∙ 2.2 - 27 = 39.6 - 27 = 12.6
3^2.2 ≈ 12.6
bobpursley
This is a homework.
No exact value is required.
That is used to practice interpolation.
To estimate the value of 3 to the power of 2.2 using linear interpolation, we need to find two known values and interpolate between them.
We know that 3 to the power of 2 is 9 and 3 to the power of 3 is 27.
First, we need to calculate the step size, which is the difference between the powers:
Step size = 3 - 2 = 1
Next, we calculate the interpolation factor (IF):
IF = 2.2 - 2 = 0.2
To find the estimated value, we can use the formula:
Estimated value = Known value + (IF * Step size)
Using the given values, we can substitute them into the formula:
Estimated value = 9 + (0.2 * 1)
Estimated value = 9 + 0.2
Estimated value = 9.2
Therefore, using linear interpolation, the estimated value of 3 to the power of 2.2 is approximately 9.2.
To estimate the value of 3 to the power of 2.2 using linear interpolation, we need to find two known values of 3 raised to a power that are close to 2.2. Then we can use those values to interpolate and approximate the value of 3 to the power of 2.2.
Let's find the values of 3 raised to the nearest lower and higher integer powers:
3^2 = 9
3^3 = 27
Now, we need to find the interpolation factor (IF) which represents the difference between the target exponent (2.2) and the lower exponent (2). The interpolation factor is calculated as:
IF = (target_exponent - lower_exponent) / (higher_exponent - lower_exponent)
IF = (2.2 - 2) / (3 - 2)
IF = 0.2
Next, we can use the interpolation formula to estimate the value:
estimate = lower_value + IF * (higher_value - lower_value)
estimate = 9 + 0.2 * (27 - 9)
estimate = 9 + 0.2 * 18
estimate = 9 + 3.6
estimate = 12.6
Therefore, using linear interpolation, the estimated value of 3 to the power of 2.2 is approximately 12.6.