A family is building a rectangular fountain in the backyard. The yard is also rectangular and measures 6x by 7x. The fountain is
2 answers:
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Answer:
x-intercept = 0.956
Step-by-step explanation:
You have the function f(x) given by:
(1)
Furthermore you have that at the point (a,f(a)) the tangent line to that point has a slope of -1.
You first derivative the function f(x):
(2)
To solve this derivative you use the following derivative formula:
For the derivative in (2) you have that b=2 and u=2x. You use the last expression in (2) and you obtain:
You equal the last result to the value of the slope of the tangent line, because the derivative of a function is also its slope.
Next, from the last equation you can calculate the value of "a", by doing x=a. Furhtermore, by applying properties of logarithms you obtain:
With this value you calculate f(a):
Next, you use the general equation of line:
for xo = a = 0.235 and yo = f(a) = 0.721:
The last is the equation of the tangent line at the point (a,f(a)).
Finally, to find the x-intercept you equal the function y to zero and calculate x:
hence, the x-intercept of the tangent line is 0.956