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Lady bird [3.3K]
3 years ago
14

The commute time to work in the U.S. Has a bell shaped distribution with a population mean of 24.4 minutes and a population stan

dard deviation of 6.5 minutes. What proportion of the population has a commute time: a) between 11.4 minutes and 37.4 minutes? (1 pt) b) less than 11.4 minutes? (1 pt) c) greater than 37.4 minutes? (1 pt)
Mathematics
1 answer:
Alexxx [7]3 years ago
5 0

Answer:

1. 0.9544

2. 0.0228

3. 0.0228

Step-by-step explanation:

The computation is shown below;

As we know that

At Normal distribution

P(X < A) = P(Z < \frac{(A - mean)}{standard deviation})

As per the question, the data provided is as follows

Mean = 24.4 minutes

Standard deviation = 6.5 minutes

Based on the above information

P(11.4 < X < 37.4) = P(X < 37.4) - P(X < 11.4)

= P(Z < (37.4 - 24.4) ÷ 6.5) - P(Z < (11.4 - 24.4) ÷ 6.5)

= P(Z < 2) - P(Z < -2)

= 0.9772 - 0.0228

= 0.9544

2. P(X < 11.4) = 0.0228

3. P(X ≥ 37.4) = 1 - P(X < 37.4)

= 1 - 0.9772

= 0.0228

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You have just opened a new nightclub, Russ' Techno Pitstop, but are unsure of how high to set the cover charge (entrance fee). O
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Answer:

a)q(p)=-15p+ +300

b)R(p)=-15p²+300p

c)C(p)=-30p+1600

d.1)P(p)=-15p²+330p-1600

d.2)p=$11

Step-by-step explanation:

(a).Before getting started, we are going to consider  the cartesian plane, where x-axis corresponds to the cover charge and y-axis corresponds to number of guests per night . There, we will locate the following coordinates according to the previous information:

(9,165)  (10,150).

Remember that  the slope-intercept form of the  linear equation is y=mx+b where m is the slope and b  is te intercept.

The slope can be calculated  by the next formula:

m=\frac{y2-y1}{x2-x1}

In this case we will assign  (9,165) as the first coordinate and (10,150) as the second one. The order does not matter. At the end we get the same value for m.

Therefore:

m=\frac{150-165}{10-9} = -15

Now, we have the slope and two poitns. According to this, and taking into account that  point-slope form of the equation of a line is y-y1=m(x-x1),   find a linear demand equation.

Any of two coordintates can be selected. In this case we will select the second one.

y-150=-15(x-10)   equation 1

Solve equation 1 for y

y=-15x+150+150

y=-15x+300

Above we had defined x as cover charge (p) and y as number of guests (q). Thus, rewrite equation 1.

q(p)=-15p+ +300.  

it equation shows the number of guests per night as a function of the cover charge,

(b). The nightly revenue can be calculated multplying the price of the cover charge by  the number of guests. So, we just have to multiply the equation 1 by p.

p*q(p)=p*(-15p +300)

R(p)=-15p²+300p   equation 2

(c) To get the cost function of the nightclub  we just have to multiply the cost of the two non-alcoholic drinks   by the number of the guests ( equation 1) and add to it  nightly overheads.

C(p)=2*(-15p+300)+1000

C(p)= -30p+600+1000

C(p)=-30p+1600  equation 3

(d) In order to find the profit in terms of the cover charge we will  subtract nightly costs ( equation 3) from nightly revenue (equation 2)

P(p)=R(p)-C(p)

P(p)=-15p²+300p - (-30p+1600)

P(p)=-15p²+300p+30p-1600

P(p)=-15p²+330p-1600

To determine that  the entrance fee we should charge for a maximum profit we are going to use the first derivative test (y'(x)=0) in order to find an extremum point.

Compute the first derivative of P(p)

P'(p)=-30p+330=0

30p=330

p=330/30

p= $11

The entrance fee that we should charge for a maximum profit is p=$11 per guest.

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