Final answer:
To estimate √47, use linear approximation by finding the square root of a nearby number like 49 and adjusting based on the derivative of the square root function.
Step-by-step explanation:
To use a linear approximation to estimate a value, you typically need a function or a set of data that can be approximated with a linear function. Without this, it's unclear what's being estimated as simply '47'. In some of the provided information, we find a reference to a least-squares regression line with an equation ŷ-173.51 + 4.83x, which can be used for linear approximations if 'x' is in the scope of the observed data.
However, based on the context of the question, it seems that you're being asked to estimate the square root of 47. We can do this by finding the square root of a nearby perfect square and use linear approximation to estimate the square root of 47. The perfect square closest to 47 is 49, which has a square root of 7. Since 47 is 2 less than 49, we can use the derivative of the square root function, which is 1/(2*sqrt(x)), to estimate the difference. At x=49, the derivative is 1/(2*7) or 1/14. This means that for every unit x decreases, the square root of x decreases by about 1/14. So, an approximation for the square root of 47 is 7 - (2/14), which simplifies to 7 - 1/7, or approximately 6.86.