Suppose you have 54 feet of fencing to enclose a rectangular dog pen. the function a = 27x – x2, where x = width, gives you the area of the dog pen in square feet. what width gives you the maximum area? what is the maximum area? round to the nearest tenth as necessary. width = 13.5 ft; area = 182.3 ft2 width = 13.5 ft; area = 546.8 ft2 width = 27 ft; area = 182.3 ft2 width = 27 ft; area = 391.5 ft2

The width of the dog pen is 13.5 ft and the area is 182.3 square feet.

Step-by-step explanation:

We have the function where is the width of the dog pen. So, if we want to obtain the maximum area, we must find the maximum of the function .

In order to accomplish this task, we must calculate the derivative  :

.

The next step is to find the critical points of  , which means to find the values of where  . This is equivalent to solve the equation . So, . In order to check if 27/2 is, in fact, a point of maximum we calculate the second derivative

.

Notice that , and the sufficient condition of the second derivative gives us that is a maximum.

In order to find the maximal value we evaluate at 27/2:

.

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}

total amount to be paid = $19.5

step-by-step explanation:

given ,

original price = $25.99

discount = 25%

discount amount = 25% of original price

discount amount = 25/100   * 25.99

discount amount = $6.49

hence , the total amount to be paid = original price - discount amount

                                                            = $25.99 - $6.49

                                                            =$19.5

total amount to be paid = $19.5