Suppose r(t)= cos(t)i+sin(t)j+(2t)k represents the position of a particle on a helix, where Z is the height of the particle abov

Let Selma's current age be s

Let Joyce's current age be j

Selma is three times as old as Joyce now

s=3j........(1)

Four years prior,

s-4=j-4

This means

Selma was four times Joyce's age

s-4=4(j-4)

s-4=4j-16

s=4j-16+4

s=4j-12.............(2)

Now substitute s in (2) using (1)

3j=4j-12

3j-4j=-12

-j=-12

Multiply by -1 on both sides

j=12

Now substitute j into (1)

s=3j

s=3(12)

s=36

Selma's current age = s = 36 years old

Joyce's current age = j = 12 years old

Answer:

To identify the critical point, we set the derivative equal to zero, resulting in:

Calculating the second derivative yields:

This indicates that t = 3.4375 marks the minimum value for the function. Substituting this back into the original function gives us:

Thus, the minimum annual income is found at t = 3.43 (between the years 2008 and 2009), with a value of 25.094

Step-by-step explanation:

In this scenario, we have the function:

Where P represents the annual net income from 2007-2011, and

t is the number of years since early 2005

To discover the lowest income, we utilize the derivative given by:

Setting this derivative to zero allows us to find the critical point, leading to:

Calculating the second derivative reveals:

Therefore, we conclude that t = 3.4375 is the minimum value, substituting into the original function results in:

Thus, the minimum annual income occurs at t = 3.43 (between 2008 and 2009) with the value being 25.094

Result:

20.19°

Detailed explanation:

Refer to the attached diagram related to the question. We need to determine <CAB

Utilizing the sine rule on triangle ABC:

cross-multiply

Thus, the angle <CAB measures 20.19°