Can someone help me please

Consider a circle with radius [tex]r[/tex] centered at some point [tex](R+r,0)[/tex] on the [tex]x[/tex]-axis. This circle has equation

[tex](x-(R+r))^2+y^2=r^2[/tex]

Revolve the region bounded by this circle across the [tex]y[/tex]-axis to get a torus. Using the shell method, the volume of the resulting torus is

[tex]\displaystyle2\pi\int_R^{R+2r}2xy\,\mathrm dx[/tex]

where [tex]2y=\sqrt{r^2-(x-(R+r))^2}-(-\sqrt{r^2-(x-(R+r))^2})=2\sqrt{r^2-(x-(R+r))^2}[/tex].

So the volume is

[tex]\displaystyle4\pi\int_R^{R+2r}x\sqrt{r^2-(x-(R+r))^2}\,\mathrm dx[/tex]

Substitute

[tex]x-(R+r)=r\sin t\implies\mathrm dx=r\cos t\,\mathrm dt[/tex]

and the integral becomes

[tex]\displaystyle4\pi r^2\int_{-\pi/2}^{\pi/2}(R+r+r\sin t)\cos^2t\,\mathrm dt[/tex]

Notice that [tex]\sin t\cos^2t[/tex] is an odd function, so the integral over [tex]\left[-\frac\pi2,\frac\pi2\right][/tex] is 0. This leaves us with

[tex]\displaystyle4\pi r^2(R+r)\int_{-\pi/2}^{\pi/2}\cos^2t\,\mathrm dt[/tex]

Write

[tex]\cos^2t=\dfrac{1+\cos(2t)}2[/tex]

so the volume is

[tex]\displaystyle2\pi r^2(R+r)\int_{-\pi/2}^{\pi/2}(1+\cos(2t))\,\mathrm dt=\boxed{2\pi^2r^2(R+r)}[/tex]

Answer:68688

Step-by-step explanation:123 Thats EZ

The income remaining after taxes from last year is $68,688.

The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.

To calculate the amount of income remaining after taxes, we need to subtract the total taxes paid from the gross income.

The total tax rate is the sum of the federal tax rate, FICA tax rate, and state tax rate.

Total tax rate = Federal tax rate + FICA tax rate + State tax rate

Total tax rate = 16.3% + 7.65% + 4.5%

Total tax rate = 28.45%

So the total amount of taxes paid is:

Total taxes = Gross income x Total tax rate

Total taxes = $96,000 x 28.45%

Total taxes = $27,312

To find the income remaining after taxes, we subtract the total taxes paid from the gross income:

Income after taxes = Gross income - Total taxes

Income after taxes = $96,000 - $27,312

Income after taxes = $68,688

Therefore, the income remaining after taxes from last year is $68,688.

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Answer:

Why do they delete your question and answers too??

Step-by-step explanation:

Step-by-step explanation:

A=8B  

A+B=180

8B+B=180°

9B=180°

B=20°

so  

angle A =8×20=160°

angle B= 20°

The requried measures of angles A and B are 160° and 20° respectively.

Orientation of the one line with respect to the horizontal or other respective line is known as a measure of orientation and this measure is known as the angle.

Here,
Angle A and angle B are supplementary angles and angle A is eight times as large as angle ​B,
The sum of the supplementary angle is 180°
A + B = 180
8B + B = 190
B = 20

Now,
A = 8B
A = 160°

Thus, the requried measure of angles A and B are 160° and 20° respectively.

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Answer:

The probability that dist(Ada,Bob)>(Ada,Cathy)  is very small as there is very large number of range to choose the product ==4.7*10^-9.

Step-by-step explanation:

Given:

Ada ,bob and cathy purchase electronics carries

Ada and bob commonly take 3 products and 7 independently.

And Cathy take 10  products on its own .

To Find:

probability that    dist(Ada,bob)>dis(Ada,Cathy)?

Solution:

Using Euclidean distance  is distance formula used in coordinate geometry simply known as Distance formula,

this problem is related to Euclidean Distance and Jaccard Similarity in Data mining.

1st calculate probability for x such that ,

[tex]3\leq x\leq 10[/tex]   as there are 3 common products.

P([tex]3\leq x\leq 10[/tex])

=[tex]\frac{7C(x-3)*990C(10-x)}{997C7}[/tex].............. where x=3,4,5....10. ..........(equation 1).

Now calculate for each term,we get

When

x=3,P(x=3)=0.95

x=4,P(x=4)=[tex]6.8*10^{-3}[/tex]

x=5,P(x=5)=[tex]4.1*10^{-5}[/tex]

x=6,P(x=6)=2.1*10^-7

x=7,P(x=7)=8.5*10^-10

x=8,P(x=8)=2.6*10^-12

x=9,P(x=9)=5.2*10^-15

x=10,P(x=10)=5.3*10^-18.

Now calculating the Euclidean distance,

It is distance between two points ,

So there are total of 2 points as Ada and bob

they have 3 products in common

and 7 independent products ,7  Ada and 7 bob

Total of 17 products .

1,2,3,4,5,6..........,16,17.

Consider each product number as distance between them ,

(Suppose 5 product and 1 product distance will be 4)

Similarly,

Suppose Ada is at 3rd number at the  3 product (as they have 3 product same.)

and bob  at product 17.

Hence when 3 products are similar distance between Ada and bob will be of 14.

Euclidean distance =[tex]\sqrt{14}[/tex].

Hence the Jaccard similarity =(Ada intersection Bob)/(Ada union bob)

=3/14

When 4 products are same means both will selected 6 and 6 independent product so that  the each one will get 10 products i.e. starting condition should remain same .

Hence now  bob will be at 16th term as it will take one more same product in between them

So no of same products will be 4,

Hence Ada will be at 4th term and bob will be at 16

So Euclidean distance =[tex]\sqrt{12}[/tex].

Similar For Next terms we can conclude as follows:

When

X=5 , dist(ada,bob)=[tex]\sqrt{10}[/tex],

X=6,dist(Ada,Bob)=[tex]\sqrt{8}[/tex]

X=7,dist(Ada,Bob)=[tex]\sqrt{6}[/tex]

X=8,dist(Ada,Bob)=[tex]\sqrt{4}[/tex]

X=9,dist(Ada,Bob)=[tex]\sqrt{2}[/tex]

X=10,dist(Ada,Bob)=[tex]\sqrt{0}[/tex].

Now for( Ada and cathy)

Here X ranges different but use same concept as above

Each term analog to the distance between them

Suppose 1st and 3rd term distance will be 2

First calculate

P([tex]1\leq x\leq 10[/tex]) as Cathy selects 10 products with no common between them.

P([tex]1\leq x\leq 10[/tex])

=[tex]\frac{10Cx*990C(10-x)}{1000C10}[/tex]..................equation (2)

Calculate for each term As x=1,2,3...8,9,10.

Hence

P(X=1)=9.23*10^-3  P(X=5)=3*10^-11     P(X=9)=3.8*10^-21

P(X=2)=8.4*10^-5   P(X=6)=1.5*10^-13   P(X=10)=3.8*10^-21

P(X=3)=6.9*10^-7   P(X=7)=6.1*10^-16

P(X=4)=4.9*10^-9   P(X=8)=1.9*10^-18

So Ada will have 10 products and Cathy will have 10 products

Namely,

1,2,3,4,5.......18,19,20.

So suppose 1 product is same between them will be ,

both will have 1 product so remaining will be 19 products.

Jaccard similarity =1/19

Distance to reach 1 to 19th product will be 18

So Euclidean distance =[tex]\sqrt{18}[/tex]

For next when they will 2 products in same remaining will be 18

Jaccard similarity =2/18

And Distance to reach  2 to 18 th product will be  16

Euclidean distance =[tex]\sqrt{16}[/tex]

Similar for  other

When

x=3 dist(Ada, Cathy)=[tex]\sqrt{14}[/tex]

x=4 dist(Ada, Cathy)=[tex]\sqrt{12}[/tex]

x=5  dist(Ada, Cathy)=[tex]\sqrt{10}[/tex]

x=6  dist(Ada, Cathy)=[tex]\sqrt{8}[/tex]

x=7  dist(Ada, Cathy)=[tex]\sqrt{6}[/tex]

x=8  dist(Ada, Cathy)=[tex]\sqrt{4}[/tex]

x=9  dist(Ada, Cathy)=[tex]\sqrt{2}[/tex]

x=10  dist(Ada, Cathy)=[tex]\sqrt{0}[/tex]

This sqrt(0) means both are holding same products hence they are at same point on the graph so distance with itself will be zero.

Now the Probability of distance of dist(Ada,Bob)>dist(Ada,cathy) will be

=multiplying both the probabilities equations (Adding each term probabilities and multiplying )

=Equation(1) *Equation( 2).

=Summation Of P(3≤x≤10)*summation of P(1≤x≤10)

=4.7*10^-9.

In larger number of product event of in large space ,it is difficult( less likely)  that they will chose same product .

This question involves complex probability in high dimensional spaces, making it difficult to provide an exact mathematical solution. Using a simulated method like Monte Carlo simulations could potentially provide an estimated answer, but it's key to remember that these are just approximations.

This question involves a complex application of probability and distance in a high dimensional space (analogous to the recommendation system in e-commerce). The Euclidean distance between Ada and Bob would be zero for the first three products. For the other seven products that Ada and Bob independently purchase, the probability that they choose the same product would influence the Euclidean distance between them. However, getting an exact mathematical model for this is quite complex. The distance between Ada and Cathy is even more complicated because Cathy is selecting products randomly from all 1000 products.

Because of the randomness and high dimensionality involved, this question may not have an exact solution but could be estimated using simulations. In settings like these, Monte Carlo simulations, which involve running many trials with randomized inputs and calculating the averages, can be useful. However, it's important to remember these are only estimates and not exact mathematical solutions.

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Answer:

The four in the hundreds place on the right is bigger because it is a whole number and the 4 behind the decimal on the left is smaller because it is a decimal.

Step-by-step explanation:

Answer:

38ft

Step-by-step explanation:

I'm assuming that you mean that the dimensions are 8ft by 11ft

If so, you need to add all the sides together, there should be two 8ft sides and two 11ft sides

8*2 = 16 (or 8+8

11*2= 22 (or 11+11

16+22 = 38

The exponential equation that represents the growth of the bacteria over time, considering the given initial and future population sizes and the time that has passed, is P(t) = 360 × e^[(ln(1000/360)/15)t]. This formula adjusts the rate of growth to match the change from 360 bacteria to 1000 bacteria over a 15-minute period.

In this situation, we are dealing with a phenomenon commonly seen in biology called exponential growth which can be modeled mathematically. For this case, we can use the exponential growth formula, P(t) = P_0 × e^(rt), where P(t) is the future population size, P_0 is the initial population size, r is the growth rate, and t is the time that has passed.

Firstly, we know the initial population size (P_0) is 360 bacteria. The population size after 15 minutes (from 5 minutes to 20 minutes is 15 minutes) is 1000 bacteria. Therefore, if we plug in these values, we have 1000 = 360 × e^(15r).

To solve for r, we start by dividing both sides by 360 which results in 1000/360 = e^(15r). Taking the natural logarithm (ln) of both sides to isolate the exponential part, we get ln(1000/360) = 15r. Finally, divide by 15 yielding r = ln(1000/360)/15.

Therefore, the exponential equation representing this scenario is P(t) = 360 × e^[(ln(1000/360)/15)t], showing the population P(t) of the bacteria at any time t.

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Answer:

Triangle F G H is similar to Triangle K L J

Step-by-step explanation:

Angle H = Angle J

Angle G = Angle L

Angle F = Angle K

KLJ is similar to FGH

If the lengths are also equal, then they're congruent

Triangle FGH is similar to triangle JKL.

To express the relationship between the two triangles using a similarity statement, we need to match corresponding angles.

Since the triangles are similar, corresponding angles are congruent.

In triangle FGH,

the angles are 65° , 24°  , and 91 °

In triangle JKL,

the angles are 24° ,91°  , and 65°

We see that the angles 24°  , 91°, and 65° in triangle JKL match the angles in triangle FGH respectively.

So, the similarity statement expressing the relationship between the two triangles is:

Triangle FGH is similar to triangle JKL.

Answer:

[tex]z=\frac{0.616-0.5}{\sqrt{\frac{0.5(1-0.5)}{297}}}=3.998[/tex]  

[tex]p_v =2*P(z>3.998)=0.0000639[/tex]  

With the most common significance levels used [tex]\alpha= 0.1, 0.05, 0.01[/tex] we see that the p value is lower than the significance level so then we have enough evidence to reject the null hypothesis and we can say that the true proportion is significantly higher than 0.5

Step-by-step explanation:

Information given  

n=297 represent the random sample of male taken

X=183 represent the  men who said yes, they had driven a car when they probably had too much alcohol

[tex]\hat p=\frac{183}{297}=0.616[/tex] estimated proportion of men who said yes, they had driven a car when they probably had too much alcohol

[tex]p_o=0.5[/tex] is the value that we want to test

z would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value (variable of interest)  

Hypothesis to test

We need to conduct a hypothesis in order to test the claim that the majority of men in the population (that is, more than half) would say that they had driven a car when they probably had too much alcohol, and the system of hypothesis are:  

Null hypothesis:[tex]p\leq 0.5[/tex]  

Alternative hypothesis:[tex]p > 0.5[/tex]  

The statistic is given by:

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

After replace we got:

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.616-0.5}{\sqrt{\frac{0.5(1-0.5)}{297}}}=3.998[/tex]  

Decision

We have a right tailed test so then the p value would be:  

[tex]p_v =2*P(z>3.998)=0.0000639[/tex]  

With the most common significance levels used [tex]\alpha= 0.1, 0.05, 0.01[/tex] we see that the p value is lower than the significance level so then we have enough evidence to reject the null hypothesis and we can say that the true proportion is significantly higher than 0.5

Answer:

The answer is A.histogram

Step-by-step explanation:

Statistical data can be represented on charts such as histograms, box plots, etc.

The (a) histogram shows the number of values within series of a range

From the question, we understand that:

The chart that supports the above highlights is the histogram.

This is so because, it can be used to illustrate grouped data (i.e. data in series of a range), while others are not suitable for grouped data.

Hence, the chart is (a) histogram

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Answer:: D(t)= 50(4/5)^t

Step-by-step explanation: If 1/5 of the temperature difference is lost each minute, that means 4/5 of the difference remains each minute. So each minute, the temperature difference is multiplied by a factor of 4/5 (or 0.8).

If we start with the initial temperature difference, 50° Celsius, and keep multiplying by 4/5, this function gives us the temperature difference t minutes after the cake was put in the cooler.

Answer:

See answer below

Step-by-step explanation:

Hi there,

The prompt is trying to showcase exponential functions, and specifically exponential decay, where over the course of the independent variable (time in this example) the dependent variable (temp difference) exponentially drops.

To start, when a math prompt says something like "at the moment xyz begins" it usually means time zero. Thus, we have 1 point already, the y-intercept:

[tex]D(0)= 50 \ C[/tex]°

Now, we notice that it says it "loses 1/5 of its original value every minute" which is code for exp. decay.  So, to account for this, the remaining value is just b = 1 - 0.2 = 0.8.

Exponential Decay formula:

[tex]f(x) = a (1-r)^x[/tex]  where a is a constant, and constant r [tex]< 1[/tex]

[tex]D(t) = 50 * (0.8)^t \ C[/tex]°    where t is in minutes

thanks,

Answer:

[tex]H_o:\mu\geq 430\\\\H_a:\mu<430[/tex]

Step-by-step explanation:

-let [tex]\mu[/tex] denote the population mean in grams.

-The claim under investigation is that the machine is underfilling the bags.

-The investigator intends to determine whether or not the population mean is less than 430 grams.

#The stated hypotheses are:

The null hypothesis(mean gram of the bags is greater or equal to 430g):

[tex]H_o:\mu\geq 430\\\\[/tex]

-Alternative hypothesis is the population mean is less than 430g:

[tex]H_a:\mu<430[/tex]

We're given the following probabilities:

[tex]P(A_1)=0.30[/tex]

[tex]P(A_2)=0.55[/tex]

[tex]P(A_1\cap A_2)=0[/tex]

[tex]P(B\mid A_1)=0.20[/tex]

[tex]P(B\mid A_2)=0.05[/tex]

(a) Yes, [tex]A_1[/tex] and [tex]A_2[/tex] are mutually exclusive. This is exactly what zero probability of their intersection means. The two events cannot occur simultaneously.

(b) Use the definition of conditional probability to expand:

[tex]P(A_1\cap B)=P(A_1)P(B\mid A_1)=0.30\cdot0.20=0.06[/tex]

[tex]P(A_2\cap B)=P(A_2)P(B\mid A_2)=0.55\cdot0.05=0.0275[/tex]

(c) By the law of total probability,

[tex]P(B)=P(A_1\cap B)+P(A_2\cap B)=0.06+0.0275=0.0875[/tex]

(d) Bayes' theorem says

[tex]P(A_1\mid B)=\dfrac{P(A_1)P(B\mid A_1)}{P(B)}=\dfrac{0.30\cdot0.20}{0.0875}\approx0.686[/tex]

[tex]P(A_2\mid B)=\dfrac{P(A_2)P(B\mid A_2)}{P(B)}=\dfrac{0.55\cdot0.05}{0.0875}\approx0.314[/tex]

Using probability concepts, it is found that:

a) Since [tex]P(A1 \cap A2) = 0[/tex], events A1 and A2 are mutually exclusive.

b) P(A1 ∩ B) = 0.06, P(A2 ∩ B) = 0.0275.

c) P(B) = 0.0875.

d) P(A1|B) = 0.6857 and P(A2|B) = 0.3143.

-----------

Conditional Probability  

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]

In which

-----------

Bayes Theorem:

Two events, A and B.

[tex]P(B|A) = \frac{P(B)*P(A|B)}{P(A)}[/tex]

In which

-----------

Item a:

-----------

Item b:

To compute these probabilities, we use conditional probability.

A1 and B:

[tex]P(B|A1) = \frac{P(A1 \cap B)}{P(A1)}[/tex]

Since [tex]P(A1) = 0.3, P(B|A1) = 0.2[/tex]

[tex]0.2 = \frac{P(A1 \cap B)}{0.3}[/tex]

[tex]P(A1 \cap B) = 0.2(0.3) = 0.06[/tex]

Thus P(A1 ∩ B) = 0.06.

A2 and B:

[tex]P(B|A2) = \frac{P(A2 \cap B)}{P(A2)}[/tex]

Since [tex]P(A2) = 0.55, P(B|A1) = 0.05[/tex]

[tex]0.05 = \frac{P(A2 \cap B)}{0.55}[/tex]

[tex]P(A2 \cap B) = 0.05(0.55) = 0.0275[/tex]

Thus P(A2 ∩ B) = 0.0275.

-----------

Item c:

P(B) can be written as:

[tex]P(B) = P(A1)P(B|A1) + P(A2)P(B|A2) = 0.3(0.2) + 0.55(0.05) = 0.06 + 0.0275 = 0.0875[/tex]

Thus P(B) = 0.0875.

-----------

Item d:

Applying Bayes Theorem, first for A1 given B.

[tex]P(A1|B) = \frac{P(A1)P(B|A1)}{P(B)} = \frac{0.3(0.2)}{0.0875} = 0.6857[/tex]

Then for A2 given B.

[tex]P(A2|B) = \frac{P(A2)P(B|A2)}{P(B)} = \frac{0.55(0.05)}{0.0875} = 0.3143[/tex]

Thus P(A1|B) = 0.6857 and P(A2|B) = 0.3143.

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Answer:

x = 4

x = -4

Step-by-step explanation:

bro i dont know but i hope this help a little, sorry

Answer:

The distance between the cars is increasing at a rate of 65miles/hour

Step-by-step explanation: Please see the attachments below

We just need to find the area of the parallelogram and the triangle, and then add.

Parallelogram: The area is base times height. So, we can write 4 * 6 = 24 in^2.

Triangle: The area is base times height divided by two. So, it's 4 * 6 / 2 = 12 in^2.

24 + 12 gives us an answer of 36 inches squared.

Answer:

36 inches squared

Step-by-step explanation:

0.000000452 in scientific notation would be 4.52 × [tex]10^{-7}[/tex]

Increasing the sample size in a statistical analysis, from n=3 to n=16, will likely lead to a decrease in the width of the confidence interval, making it more precise. Therefore, the most likely confidence interval after the change would be option C, 67.5 to 72.5.

In statistics, increasing the sample size reduces the standard error of the mean. The standard error defines the width of the confidence interval for a given sample size. If we increase the sample size from n=3 to n=16, the standard error and thus the width of the confidence interval will likely decrease, assuming all else stays the same. Hence, the confidence interval will be more precise.

Considering the given options, option C, a confidence interval of 67.5 to 72.5, represents a narrower interval and is therefore the most likely result after increasing the sample size from n=3 to n=16.

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Answer:

The correct option is (A).

Step-by-step explanation:

The (1 - α)% confidence interval for difference in proportion formula is,

[tex]CI=(\hat p_{1}-\hat p_{2})\pm z_{\alpha /2}\sqrt{\frac{\hat p_{1}(1-\hat p_{1})}{n_{1}}+\frac{\hat p_{2}(1-\hat p_{2})}{n_{2}}}[/tex]

The given information is:

n₁ = n₂ = 200,

X₁ = 1062,

X₂ = 900.

Compute the sample proportion as follows:

[tex]\hat p_{1}=\frac{X_{1}}{n_{1}}=\frac{1062}{2000}=0.531\\\\\hat p_{2}=\frac{X_{2}}{n_{2}}=\frac{900}{2000}=0.45[/tex]

For the 95% confidence level, the z-value is,

z₀.₀₂₅ = 1.96

*Use a z-table.

Compute the 95% confidence interval for the difference in proportion as follows:

[tex]CI=(\hat p_{1}-\hat p_{2})\pm z_{\alpha /2}\sqrt{\frac{\hat p_{1}(1-\hat p_{1})}{n_{1}}+\frac{\hat p_{2}(1-\hat p_{2})}{n_{2}}}[/tex]

     [tex]=(0.531-0.45)\pm1.96\sqrt{\frac{0.531(1-0.531)}{2000}+\frac{0.45(1-0.45)}{2000}}[/tex]

     [tex]=0.081\pm 0.031\\=(0.050, 0.112)[/tex]

Thus, the 95% confidence interval for the difference in proportion of registered voters that support this candidate between the northern and southern halves of the state is (0.050, 0.112).

The correct option is (A).

Answer:

a and c

Step-by-step explanation:

The statements that are true are:

The area of the base of the pyramid, B, is 48 in.2

A square prism with dimensions 8 inches by 8 inches by 6 inches will have a volume three times as large as this pyramid.

The volume of the pyramid is 384 in.3

A pyramid is a structure where outer surfaces are triangular and converge to a point at the top.

The volume of a pyramid with a square base is given as:

Volume = 1/3 x base area x height

We have,

The area of the base of the pyramid is given by B = (1/2)bh, where b is the length of the base and h is the height of the pyramid.

We are not given the height of the pyramid, so we cannot determine the area of the base.

Therefore, the statement "The area of the base of the pyramid, B, is 64 in.2" is not true.

The area of the base of the pyramid is given as 48 in.2 in the problem statement. Therefore, the statement "

The area of the base of the pyramid, B, is 48 in.2" is true.

The volume of a square prism with dimensions 8 inches by 8 inches by 6 inches is V = lwh = 8 x 8 x 6 = 384 in.3.

The volume of the pyramid is also given as 384 in.3 in the problem statement.

Therefore, the statement "A square prism with dimensions 8 inches by 8 inches by 6 inches will have a volume three times as large as this pyramid" is true.

The correct statement is "A square prism with dimensions 8 inches by 8 inches by 6 inches will have a volume one-third the volume of this pyramid."

This statement is not true, since the volume of the pyramid is 384 in.3 and one-third of that volume is 128 in.3, which is not the volume of the prism given in the problem. Therefore, the statement is false.

Thus,

The statements that are true are:

The area of the base of the pyramid, B, is 48 in.2

A square prism with dimensions 8 inches by 8 inches by 6 inches will have a volume three times as large as this pyramid.

The volume of the pyramid is 384 in.3

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Answer:

Null Hypothesis, [tex]H_0[/tex] : [tex]\mu \leq[/tex] 4.75  

Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] > 4.75  

Step-by-step explanation:

We are given that 260 people are randomly chosen at a shopping mall to taste-test a new brand of fruit drink. The results of the survey reveal that the average rating is 5.22 with a standard deviation of 2.31.

The marketing division of the fruit drink distributor is only interested in selling this drink if the true mean rating is more than 4.75.

Let [tex]\mu[/tex] = true mean rating.

So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu \leq[/tex] 4.75  

Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] > 4.75  

Here, null hypothesis states that the true mean rating is less than or equal to 4.75.

On the other hand, alternate hypothesis states that the true mean rating is more than 4.75.

Also, The test statistics that will be used here is One-sample t test statistics as we don't know about population standard deviation;

                                T.S.  = [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex]  ~ [tex]t_n_-_1[/tex]

Hence, the alternative hypothesis for testing whether the fruit drink distributor should sell this drink is  [tex]\mu[/tex] > 4.75.

Answer:

the answer will be 15

Step-by-step explanation:

If it took her 52 minuets in total, then 7 questions she solved.

Two or more expressions with an Equal sign is called as Equation.

Let x be number of equations.

From the given information, we know that Kate took 3 minutes to solve each equation, so the total time she spent on equations is 3x minutes.

We also know that it took her 7 minutes to solve the word problem.

So, the total time she spent on equations and the word problem is:

3x + 7

According to the problem, this total time is 52 minutes:

3x + 7 = 52

Subtracting 7 from both sides, we get:

3x = 45

Dividing both sides by 3, we get:

x = 15

Therefore, Kate solved 15 equations.

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Answer:

-1r

Step-by-step explanation:

The value for the expression -10r + 9r is -1r.

A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them.

Variables are the name given to these symbols because they lack set values.

In order to determine the values, these symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations.

Given:

-10r+9r

Now, perform operations to solve

=-10 r + 9r

= r( -10 + 9)

Using properties of integers

(-) x (+ )= -

So,

= r( -1)

= -1r

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Answer:

[tex]P(X\geq 3.4)=0.0228[/tex]

Step-by-step explanation:

Given the mean is 3.2, standard deviation is 0.8 and the sample size is 64.

-We calculate the  probability of a mean of 3.4 as follows:

#First determine the z-value:

[tex]z=\frac{\bar X -\mu}{\sigma/\sqrt{n}}\\\\=\frac{3.4-3.2}{0.8/\sqrt{64}}\\\\=2.000[/tex]

#We then determine the corresponding probability on the z tables:

[tex]Z(X\geq 3.4)=1-P(X<3.4)\\\\=1-0.97725\\\\=0.0228[/tex]

Hence, the probability of obtaining a sample mean this large or​ larger is 0.0228

To find the probability of obtaining a sample mean of 3.4 pounds or larger, calculate the z-score and find the corresponding probability using the standard normal distribution table.

To find the probability of obtaining a sample mean of 3.4 pounds or larger, we need to calculate the z-score for the sample mean and then find the corresponding probability using the standard normal distribution table.

First, calculate the z-score using the formula: z = (x - μ) / (σ / √n), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Substituting the given values, we have: z = (3.4 - 3.2) / (0.8 / √64) = 0.2 / (0.8 / 8) = 0.2 / 0.1 = 2.

Next, we can find the probability by looking up the z-score of 2 in the standard normal distribution table. The probability of obtaining a sample mean of 3.4 pounds or larger is approximately 0.0228 or rounded to four decimal places.

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The equivalent expression to log(2) - log(6) is D. log(1/3), since subtraction in logarithms indicates division, making the expression log(2/6) which simplifies to log(1/3).

The question asks which expressions are equivalent to log(2) - log(6). According to the properties of logarithms, the logarithm of a number resulting from the division of two numbers is the difference between the logarithms of the two numbers (log a - log b = log(a/b)). Therefore, log(2) - log(6) = log(2/6) = log(1/3).

Hence, the only equivalent expression from the given options is D. log(1/3). Option A is incorrect because log(2) + log(1/6) would imply multiplication, not division. Option B is just log(2) without subtraction. Option C, log(3), does not represent the given difference.

To find the length of the sides of squares cut from a piece of tin to create an open box, we use the dimensions of the tin and the area of the box's base to set up and solve a quadratic equation.

The student is asking for the length of the sides of the squares that are cut out from a piece of tin to create an open box. Given that the tin measures 30 cm by 70 cm, and the area of the base of the resulting box is 1536 cm2, we can set up an equation to solve for the side length of the squares.

Let's denote the side length of the squares as x. After the squares are cut out, the length and width of the base of the box will be (70 - 2x) and (30 - 2x) respectively. The area of the base is given by:

Area = length × width
1536 cm2 = (70 - 2x)(30 - 2x)

By expanding this and solving the quadratic equation for x, we can find the length of the sides of the squares cut from each corner of the tin.

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Answer:

The number of false positives is 108. The right answer is A

Step-by-step explanation:

According to the given data we have the following:

probability of negative results which are correcly identified =0.7

probability of negative results which are wrongly identified =1-0.7 =0.3

hence, probability of negative result = 1-0.1 =0.9

Therefore, in order to calculate the number of false positives we would have to use the following formula:

false positives =total number * probability of negative results which are wrongly identified * probability of negative result=400 * 0.9*(1-0.7) = 108

false positives = 400 * 0.9*(1-0.7)

false positives = 108

The number of false positives is 108

The correct answer is c. 132 oz. To convert the cat's weight of 8 1÷4 pounds to ounces, multiply by 16.

Converting Pounds to Ounces

The cat weighs 8 1/4 pounds. To find out how many ounces this is, we need to convert pounds to ounces. We know from the conversion rate that 1 pound equals 16 ounces.

Step-by-Step Calculation

Thus, the cat weighs 132 ounces, so the correct answer is c. 132 oz.