A coral reef is 2 miles due south of its closest point along a straight shoreline. A visitor is staying in a cabin that is 4 miles west of that point. The visitor is planning to go from the cabin to the coral reef. Suppose the visitor runs at a rate of 6 mph and swims at a rate of 3 mph. How far should the visitor run to minimize the time it takes to reach the coral reef

5.54 miles

Step-by-step explanation:

1. Assume she runs up to a point that is x miles to the east of the coral reef:

distance running =

distance swimming =

2. Times:

time= distance / speed

time running =

time swimming =

3. Total time

time running + time swimming:  

4. Minimize time

The minimum time is when the derivative of the function of time is zero.

Find the derivative of T(x), using rule of the sum, the derivative of the sum is the sum of the derivatives, and the chain rule:

Solve for x:

5. Distance the visitor should run to minimize the time

Hence, the visitor should run 5.54 miles.

please mark me brainliest!

1. 3

2. 4

3. 1

hope this ! have a great day.

-wfz

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range: