Find area of circle

gulaghasi [49]

11. The sum S of the first n consecutive positive integers, 1, 2, 3, ... , n, is given by the formula S=0.5n(n+1). Find n when S=120 and check your answer by addition.

$120 = \frac 1 2 n (n+1)$

$240 = n^2 + n$

$n^2 + n -240 = 0$

We need two numbers whose sum is 1 and multiply to -240, so two numbers whose difference is 1 and multiply to 240, 15 and 16. We factor:

$(n+16)(n-15)= 0$

$n=-16 \textrm{ or } n=15$

We reject the negative root.

n=15

Check:

1+2+3+4+5+6+7+8+9+10+11+12+13+14+15

=1+15 + 2+14 + 3+13 + 4+12 + 5+11 +6+10 + 7+9 + 8

= 7(16)+8

= 120 good

13. A boat travels downstream (with the current) for 36 miles and then makes the return trip upstream (against the current). The trip downstream took 3/4 of an hour less than the upstream trip. If the rate of the current is 4 mph, find the rate of the boat in still water and the time for each part of the trip.

Let v be the speed of the boat in still water.

The speed downstream is v+4 so the time to travel 36 miles downstream is 36/(v+4)

The speed upstream is v-4 so the time to travel 36 miles upstream is 36/(v-4)

Writing our equation,

3/4 = 36/(v-4) - 36/(v+4)

3 = 144 (1/(v-4) - 1/(v+4))

3(v-4)(v+4) = 144( (v+4) - (v-4) )

v^2 - 16 = 48(8)

v^2 = 48(8) + 16 = 400

v = 20 mph, the rate of the boat in still water

Time downstream is 36/(20+4)=36/24 = 3/2 = 1.5 hours

Time upstream is 36/(20-4)=36/16=9/4 = 2.25 hours

3/4 hour difference, good

5 0

3 years ago

Principal=2000,R=12percent,Time=2years and 6 months.

almond37 [142]

Principal=P=2000
Rate of interest=12%
Time=2years and 6months=2.5years
Interest=I

$\boxed{\sf I=\dfrac{PRT}{100}}$

$\\ \sf\longmapsto I=\dfrac{2000(12)(2.5)}{100}$

$\\ \sf\longmapsto I=\dfrac{24000(2.5)}{100}$

$\\ \sf\longmapsto I=\dfrac{60000}{100}$

$\\ \sf\longmapsto I=600$

7 0

3 years ago

Read 2 more answers

You have a garden ABCD that you decide to move to another

o-na [289]

Answer: no

Step-by-step explanation: no

8 0

3 years ago

Read 2 more answers

A marketing consultant observed 50 consecutive shoppers at a supermarket. One variable of interest was how much each shopper spe

Diano4ka-milaya [45]

Answer:

a. $\\ \mu = USD\;51.34$ ; b. The 95% interval is USD [45.82, 56.86].

Step-by-step explanation:

The mean amount a customer spends at the grocery store

The mean is the sum of all of the data divided by the number of observations for this case, that is, n = 50.

The sum of all of the data is: 5.24+11.97+13.59+15.52+15.69+19.96+20.54+22.59+27.25+27.29+28.25+30.00+31.63+33.01+37.21+40.36+40.90+42.83+45.66+46.67+47.49+49.32+50.12+51.71+52.2 1+53.57+56.16+56.36+56.88+58.65+59.42+60.20+60.48+62.23+62.50+67.95+70.45+73.08+74.15+74.48+74.78+75.75+76.27+76.47+78.92+84.09+84.91+85.06+88.45+88.91 = 2567.17.

The mean $\\ \mu$ is

$\\ \mu = \frac{2567.17}{50}$

$\\ \mu = \frac{2567.17}{50} = 51.34$

Then, the mean amount a customer spends at the grocery store is USD 51.34.

The 95% confidence interval for the mean amount spent

For the 95% confidence interval for the mean, we have the value for the mean = 51.34, as we have just found, the value for the standard deviation is already given as USD 20.00. There are 50 observations, so n = 50.

Moreover, for a 95% confidence, the value for the corresponding z-score is z = 1.96 (this is the confidence coefficient). Then, the formula for finding the "limits" values for the confidence interval is as follows:

For the upper limit

$\\ \overline{X} + 1.96*\frac{\sigma}{\sqrt{n}}$

$\\ 51.34 + 1.96*\frac{20}{\sqrt{50}}$

$\\ 51.34 + 5.52$

$\\ 56.86$

For the lower limit

$\\ \overline{X} - 1.96*\frac{\sigma}{\sqrt{n}}$

$\\ 51.34 - 1.96*\frac{20}{\sqrt{50}}$

$\\ 51.34 - 5.52$

$\\ 45.82$

Thus

The 95% confidence is USD [45.82, 56.86].

Mathematically

$\\ P(45.82 \leq \mu \leq 56.86) = 0.95$

There is a probability of 0.95 that the values are between [45.82, 56.86], both inclusive.

We have to remember here that we are dealing with a sample. This sample has a standard deviation, called standard error of the sample, and is $\\ \frac{20}{\sqrt{50}} \approx 2.83$ .