Initially, there were only 86 weeds in the garden. the weeds grew at a rate of 29% each week. the following function represents the weekly weed growth: f(x) = 86(1.29)x. rewrite the function to show how quickly the weeds grow each day.
a. f(x) = 86(1.04)x; grows approximately at a rate of 0.4% daily
b. f(x) = 86(1.04)7x; grows approximately at a rate of 4% daily
c. f(x) = 86(1.29)7x; grows approximately at a rate of 20% daily
d. f(x) = 86(1.297)x; grows approximately at a rate of 2% daily

The exponential function which shows the daily growth rate of the weed is

The growth function for the growth rate per week is given as :

An exponential function is generally written thus :

The growth rate per week, r = 29% = 0.29

The growth rate per day can be calculated thus :

Rewriting the function :

Hence, 4% represents the daily growth rate of the weed.

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Step-by-step

The base of the exponential function is 1.29 for 7 days, as in

f(x) = 86*(1.29)^x

The new rate for days can be calculated by dividing x by 7 (where x remains the number of weeks), namely

f(x) = 86*1.29^(x/7)

Using the law of exponents, b^(x/a) = b^(x*(1/a)) = (b^(1/a))^x

we simplify by putting b=1.29, a=7 to get

f(x) = 86*(1.29^(1/7))^x

f(x) = 86*(1.037)^x      since 1.29^(1/7) evaluates to 1.037

Rounding 1.037 to 1.04 we get a (VERY) approximate function

f(x) = 86 * (1.04^x)

1.04 is very approximate because 1.04^7 is supposed to get back 1.29, but it is actually 1.316, while 1.037^7 gives 1.2896, much closer to 1.29.