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2 years ago

The base diameter and the height of a cone are both equal to x units.

Mathematics

1 answer:

Vera_Pavlovna [14]2 years ago

6 0

Answer:

π x³ /12 is the volume of the cone if A cone has a height of x and a diameter of x

Step-by-step explanation:

The base diameter and the height of a cone are both equal to x units

Diameter of cone = x units

Radius = Diameter /2

=> Radius of cone = x/2 units

Height of cone =x units

Volume of cone = (1/3) π r²h

r = radius h = height

=> Volume of cone = (1/3)π (x/2)² * x

= (1/3) π (x²/4) * x

= (1/12) π x³

= π x³ /12 cubic units

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The following data represent the pH of rain for a random sample of 12 rain dates. A normal probability plot suggests the data co

pshichka [43]

Answer:

(1) The point estimate for the population mean is 4.925.

(2) Therefore, a 95% confidence interval for the population mean pH of rainwater is [4.715, 5.135] .

(3) Therefore, a 99% confidence interval for the population mean pH of rainwater is [4.629, 5.221] .

(4) As the level of confidence increases, the width of the interval increases.

Step-by-step explanation:

We are given that the following data represent the pH of rain for a random sample of 12 rain dates.

X = 5.20, 5.02, 4.87, 5.72, 4.57, 4.76, 4.99, 4.74, 4.56, 4.80, 5.19, 4.68.

(1) The point estimate for the population mean is given by;

Point estimate, $\bar X$ = $\frac{\sum X}{n}$

= $\frac{5.20+5.02+ 4.87+5.72+ 4.57+ 4.76+4.99+ 4.74+ 4.56+ 4.80+5.19+ 4.68}{12}$

= $\frac{59.1}{12}$ = 4.925

(2) Let $\mu$ = mean pH of rainwater

Firstly, the pivotal quantity for finding the confidence interval for the population mean is given by;

P.Q. = $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$ ~ $t_n_-_1$

where, $\bar X$ = sample mean = 4.925

s = sample standard deviation = 0.33

n = sample of rain dates = 12

$\mu$ = population mean pH of rainwater

Here for constructing a 95% confidence interval we have used a One-sample t-test statistics as we don't know about population standard deviation.

So, 95% confidence interval for the population mean, $\mu$ is ;

P(-2.201 < $t_1_1$ < 2.201) = 0.95 {As the critical value of t at 11 degrees of

freedom are -2.201 & 2.201 with P = 2.5%}

P(-2.201 < $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$ < 2.201) = 0.95

P( $-2.201 \times {\frac{s}{\sqrt{n} } }$ < ${\bar X-\mu}$ < $2.201 \times {\frac{s}{\sqrt{n} } }$ ) = 0.95

P( $\bar X-2.201 \times {\frac{s}{\sqrt{n} } }$ < $\mu$ < $\bar X+2.201 \times {\frac{s}{\sqrt{n} } }$ ) = 0.95

95% confidence interval for $\mu$ = [ $\bar X-2.201 \times {\frac{s}{\sqrt{n} } }$ , $\bar X+2.201 \times {\frac{s}{\sqrt{n} } }$ ]

= [ $4.925-2.201 \times {\frac{0.33}{\sqrt{12} } }$ , $4.925+2.201 \times {\frac{0.33}{\sqrt{12} } }$ ]

= [4.715, 5.135]

Therefore, a 95% confidence interval for the population mean pH of rainwater is [4.715, 5.135] .

The interpretation of the above confidence interval is that we are 95% confident that the population mean pH of rainwater is between 4.715 & 5.135.

(3) Now, 99% confidence interval for the population mean, $\mu$ is ;

P(-3.106 < $t_1_1$ < 3.106) = 0.99 {As the critical value of t at 11 degrees of

freedom are -3.106 & 3.106 with P = 0.5%}

P(-3.106 < $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$ < 3.106) = 0.99

P( $-3.106 \times {\frac{s}{\sqrt{n} } }$ < ${\bar X-\mu}$ < $3.106 \times {\frac{s}{\sqrt{n} } }$ ) = 0.99

P( $\bar X-3.106 \times {\frac{s}{\sqrt{n} } }$ < $\mu$ < $\bar X+3.106 \times {\frac{s}{\sqrt{n} } }$ ) = 0.99

99% confidence interval for $\mu$ = [ $\bar X-3.106 \times {\frac{s}{\sqrt{n} } }$ , $\bar X+3.106 \times {\frac{s}{\sqrt{n} } }$ ]

= [ $4.925-3.106 \times {\frac{0.33}{\sqrt{12} } }$ , $4.925+3.106 \times {\frac{0.33}{\sqrt{12} } }$ ]

= [4.629, 5.221]

Therefore, a 99% confidence interval for the population mean pH of rainwater is [4.629, 5.221] .

The interpretation of the above confidence interval is that we are 99% confident that the population mean pH of rainwater is between 4.629 & 5.221.

(4) As the level of confidence increases, the width of the interval increases as we can see above that the 99% confidence interval is wider as compared to the 95% confidence interval.

4 0

4 years ago

Which equation is the equation of a line that is perpendicular to x + 2y = 16?

gogolik [260]

Answer:

For the equation given:

$x + 2y = 16 \\ 2y = - x + 16 \\ y = - \frac{1}{2} x + 8$

for perpendicular lines, the relationship between their slopes is given by:

$m _{1} \times m _{2} = - 1$

m1 is slope for one line, and m2 is slope for another line.

$- \frac{1}{2} \times m_{2} = - 1 \\ \\ m_{2} = - 1 \times - 2 \\ { \underline{ \: \: m_{2} = 2 \: \: }}$

Therefore, equation is;

${ \boxed{ \boxed{y = 2x - 2}}}$

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3 years ago

The volume of a rectangular pyramid is 3 cubic inches. What is the volume of a rectangular prism with the same length, width, an

kap26 [50]

Answer:

Volume is the amount of 3-dimensional space an object occupies. Volume is measured in cubic units.

Step-by-step explanation:

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3 years ago

10 points for answering! What is the answer to 10, 11, and 12? Please help

Alisiya [41]

Answer:

Refer to the attachment. :)

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3 years ago

Where r is the radius of the cone's base and h is the height of the cone. Find the approximate volume of a

postnew [5]

Answer:

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(I'm assuming that 'his' is height?)

8 0

3 years ago