all the numbers from 1 to a 99 are multiplied together.Use a pattern to determine the last digit of the product
Answers
The last digit of the product of numbers from 0 to 99 is 0.
What is multiplication?Multiplication is a method of finding the product of two or more numbers.
We have,
1, 2, 3, .............till 99.
We know that any number multiplied by 10 will give us 0 as the last digit.
And there are 10, 20, 30, 40, 50, 60, 70, 80, and 90.
We also know that 5, 15, 25, 35, 45, 65, 75, 85, 95 if multiplied with an even number the last digit is zero.
So if all the numbers from 1 to 99 are multiplied together, the last digit of the product will be a zero.
Thus the last digit of the product of numbers from 0 to 99 is 0.
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Related Questions
According to a leasing firm's reports, the mean number of miles driven annually in its leased cars is 12,520 miles with a standard deviation of 2940 miles. The company recently starting using new contracts which require customers to have the cars serviced at their own expense. The company's owner believes the mean number of miles driven annually under the new contracts, μ, is less than 12,520 miles. He takes a random sample of 70 cars under the new contracts. The cars in the sample had a mean of 12,179 annual miles driven. Is there support for the claim, at the 0.05 level of significance, that the population mean number of miles driven annually by cars under the new contracts, is less than 12,520 miles? Assume that the population standard deviation of miles annually was not affected by the change to the contracts. Perform a one-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places, and round your responses as specified below. (If necessary, consult a list of formulas.) (a) State the null hypothesis H0
and the alternative hypothesis H1
. (e) Can we support the claim that the population mean number of miles driven annually by cars under the new contracts is less than 12,520 miles? Yes No \begin{tabular}{l|l|} \hline (Choose one) ∇ \\ \hlineZ \\ \hline \\ Chi-square \\ F \\ \hline \end{tabular}
Answers
This is a hypothesis testing problem where we need to determine if there is enough evidence to support the claim that the population mean number of miles driven annually
(a) The null hypothesis (H0) states that the mean number of miles driven annually under the new contracts is equal to or greater than 12,520 miles. The alternative hypothesis (H1) states that the mean number of miles driven annually under the new contracts is less than 12,520 miles.
H0: μ ≥ 12,520
H1: μ < 12,520
(e) To determine if we can support the claim that the population mean number of miles driven annually by cars under the new contracts is less than 12,520 miles, we need to perform a hypothesis test. Given that we have a random sample of 70 cars and the population standard deviation is not affected by the change in contracts, we can use the z-test.
We calculate the test statistic (z-score) using the formula:
z = (sample mean - population mean) / (population standard deviation / √sample size)
Substituting the values from the problem, we get:
z = (12,179 - 12,520) / (2940 / √70)
By calculating the z-value, we can compare it to the critical value from the standard normal distribution at a significance level of 0.05. If the z-value falls in the rejection region (less than the critical value), we can reject the null hypothesis and support the claim.
In this case, since we are performing a one-tailed test and want to determine if the population mean is less than 12,520, we look for the critical value corresponding to the 0.05 level of significance in the left tail of the standard normal distribution.
If the calculated z-value is less than the critical value, we can support the claim that the population mean number of miles driven annually by cars under the new contracts is less than 12,520 miles. If the calculated z-value is greater than the critical value, we do not have enough evidence to support the claim.
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"Write 1/100 using a negative exponent other than -1. Answer: 10^-2." Why is 10^-2 the answer?
Answers
Step-by-step explanation:
\( \frac{1}{100} = \frac{1}{ {10}^{2} } = {10}^{ - 2} \\ \\ reason \\ \: \because \: \frac{1}{ {a}^{2} } = {a}^{ - 2} \\ \\ \implies \: \frac{1}{ {10}^{2} } = {10}^{ - 2} \)
i dont know if what i wrote was right but someone help
Answers
Answer:
x = 12/7
Step-by-step explanation:
You put: What you should have had is:
13x - 3y = 12 13x - 3y = 12
- x + 3y = 12 + x + 3y = 12
Now that we have that, we can do
14x = 24
Than, divide 24 by 14
x = 24/14
which we can simplify by dividing both by 2
x = 12/7
We can't simplify it down anymore, so this is our answer.
HELP PLZ It is your turn to wash the dishes. When you are finished, you begin to drain the water from the sink. The water's draining can be defined by the function V = -.75t + 10 where V is the volume of water in gallons and t is the time in minutes. What is the rate of change and would you interpret that?
Answers
Answer:
The rate of change is -0.75
Step-by-step explanation:
For each unit of t the volume of water is decreasing by 0.75.
At t=13.333 the volume of water is 0. at t=0 the volume of water is 10
Hope this helps
Factor completely.
4x² - 4x + 1
Answers
if the post office and the library are 1 mile apart, which building is 5 6 mile from the post office?
Answers
Answer:
there is not enough information to answer the question corrcectly, post the question again and dont be not-smart this time
Step-by-step explanation:
Which of the following would be a sub-heading in an outline about pets?
1. dogs
2. shoes
3. flowers
4. rocks
Answers
if a 10.0 ml sample of vinegar is titrated with 24.30 ml of 0.231 M NaOH to reach a pale pink endpoint. Calculate the molarity of the acetic acid in the vinegar, and the mass percent of acetic acid in the vinegar, assuming the vinegar solution has a density of 1.01 g/ml. Show all work
Answers
The molarity of acetic acid in the vinegar is 0.561 M and the mass percent of acetic acid in the vinegar is 33.6%.
In this problem, we are given the volume and concentration of the base (NaOH) used to titrate a sample of vinegar, and we need to find the molarity of acetic acid in the vinegar and the mass percent of acetic acid.
To start, we can use the balanced chemical equation for the reaction between acetic acid and NaOH to find the moles of NaOH used in the titration:
CH₃COOH + NaOH → CH₃COONa + H₂O
From the equation, we can see that the ratio of moles of acetic acid to moles of NaOH is 1:1. Therefore, the number of moles of NaOH used is the same as the number of moles of acetic acid in the original vinegar sample.
Molarity of acetic acid in the vinegar:
moles of NaOH = concentration of NaOH × volume of NaOH used
moles of acetic acid = moles of NaOH
molarity of acetic acid = moles of acetic acid / volume of vinegar sample
Substituting the given values, we get:
moles of NaOH = 0.231 M × 0.02430 L = 0.00561 moles
moles of acetic acid = 0.00561 moles
molarity of acetic acid = 0.00561 moles / 0.0100 L = 0.561 M
Mass percent of acetic acid in the vinegar:
mass of acetic acid = density × volume × molarity × molecular weight
mass percent = (mass of acetic acid / mass of vinegar sample) × 100%
Substituting the given values and the molecular weight of acetic acid (60.05 g/mol), we get:
mass of acetic acid = 1.01 g/mL × 0.0100 L × 0.561 mol/L × 60.05 g/mol = 3.39 g
mass percent = (3.39 g / (1.01 g/mL × 0.0100 L)) × 100% = 33.6%
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8x - 3 /AX 4x + 3 When angles form a linear pair, their sum is 180°. 8x - 3+ 4x + 3 = 180 [?]x + [] = 180
![8x - 3 /AX 4x + 3 When angles form a linear pair, their sum is 180. 8x - 3+ 4x + 3 = 180 [?]x + [] =](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/qyHHRwzU1NH60DvaXnSWaKqNVEUte8Mf.jpeg)
Answers
Answer:
12x + 0 = 180x = 15Step-by-step explanation:
You are given the equation 8x -3 +4x +3 = 180 and asked to simplify it and solve for x.
SimplifiedCollecting terms, we have ...
(8 +4)x +(-3 +3) = 180
12x + 0 = 180
Dividing by the coefficient of x gives ...
x = 180/12 = 15
The value of x is 15.
using a Binomial Distribution calculator
I receive about 20 ± 3 (n = 40) robo-calls per week. (note that 3 is the standard deviation based on 40 weeks) (a) Assuming that the calls are random, what is the probability that I'll receive exactly 15 robo-calls next week? (b) What is the probability that I'll receive fewer than 15 calls next week? (c) Next month, how likely am I to receive fewer than 60 calls? (e) During the first week of July, I received only 12 calls. Should I assume that the callers left for the 4th of July holiday, or is an extreme value such as 12 calls likely based on chance alone? (f) Based on data I collected over 40 weeks, do the calls appear to be aggregated, random, or evenly distributed? Calculate a statistic to support this answer
Answers
a) The probability that I'll receive exactly 15 robo-calls next week 0.323386.
b) The probability of receiving exactly 15 robo-calls next week, assuming the calls are random, is 0.078145 or 7.81%.
c) The probability of receiving fewer than 60 robo-calls next month, is 0.0004 or 0.04%.
e) It is reasonable to consider other factors such as the 4th of July holiday or other external influences impacting the number of robo-calls received during that week.
f) It appears that the calls are more likely to be randomly distributed or possibly evenly distributed, rather than aggregated.
Using binomial distribution formula
P(X = k) = C(n, k) \(p^k (1 - p)^{(n - k)\)
where:
- P(X = k) is the probability of getting exactly k successes (k robo-calls in this case),
- n is the number of trials (weeks),
- p is the probability of success (probability of receiving a robo-call).
In this case, n = 40 (weeks), and the average number of robo-calls received per week is 20 with a standard deviation of 3.
To calculate the probability, we need to convert the average and standard deviation to the probability of success (p). We can do this by dividing the average by the number of trials:
p = average / n = 20 / 40 = 0.5
Now we can substitute the values into the binomial distribution formula:
P(X = 15) = C(40, 15) *\((0.5)^{15} (1 - 0.5)^{(40 - 15)\)
P(X = 15) = 3,342,988 x 0.0000305176 x 0.0000305176
= 0.323386
b) The probability that I'll receive fewer than 15 calls next week
P(X = 15) = C(40, 15) \((p)^{15} (1 - p)^{(40 - 15)\)
P(X = 15) = 847,660 x 0.0000305176 x 0.0000305176
= 0.078145
Therefore, the probability of receiving exactly 15 robo-calls next week, assuming the calls are random, is 0.078145 or 7.81%.
(c) P(X < 60) = P(Z < (60 - 80) / 6)
= P(Z < -20 / 6)
= P(Z < -3.33)
Therefore, the probability of receiving fewer than 60 robo-calls next month, assuming the average and standard deviation per week hold, is 0.0004 or 0.04%.
e) In this case, since the z-score is -2.67, which falls outside the range of -1.96 to 1.96, we can conclude that receiving only 12 calls during the first week of July is statistically significant.
It suggests that the observed value is unlikely to occur based on chance alone, and it is reasonable to consider other factors such as the 4th of July holiday or other external influences impacting the number of robo-calls received during that week.
f) It appears that the calls are more likely to be randomly distributed or possibly evenly distributed, rather than aggregated.
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Please helpp i need this rn if possible
Answers
Answer:
140, 60
Step-by-step explanation:
15x+5 + < CDB =180 line form an 180 angle
180-80-6x-6 = < CDB sum of angles in a triangle is 180
substitute < CDB from second to first
15x+5+180 -80-6x-6 =180
combine like terms
9x -81 =0
x=81/9=9
so angle YDC = 15*9+5 = 140 degrees
and angle DBC = 6*9+6= 60 degrees
Can someone answer this correctly?
Answers
The exponential function f(x) is defined as follows:
f(x) = 5(2)^x.
How to define the exponential function?An exponential function is defined as follows:
y = a(b)^x.
In which:
a is the value of y when x = 0.b is the rate of change.From the table, when x = 0, f(x) = 5, hence the parameter a is given as follows:
a = 5.
When x is increased by one, f(x) is multiplied by 2, hence the parameter b is given as follows:
b = 2.
Meaning that the function is defined as follows:
f(x) = 5(2)^x.
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find the increase in length,area or volume with:o.5m, 10(linear expansion),15(rise in temp),
Answers
Answer:
Bonjour is the bonjour of qui
Step-by-step explanation:
Points points
a census reports that the mean retirement age is 68.3 years. in a random sample, the mean retirement age is 65.8 years. what is the mean of 68.3 years?
Answers
The mean retirement age in the census report is 68.3 years.
The given information states that the population mean retirement age is 68.3 years, and a random sample of retirement age has a sample mean of 65.8 years. We can use this information to estimate the population mean with a certain level of confidence.
However, the question asks us to find the mean of 68.3 years, which is simply the given population mean. Therefore, we can state that the mean of 68.3 years remains the same, as it is not affected by the sample mean or any other sample statistic.
In other words, the population mean of 68.3 years is a fixed value, and it does not change based on the sample mean or any other sample statistic. Therefore, we can simply state that the mean retirement age is 68.3 years, which is the given information provided in the question.
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You buy a 1:1000 cale model of the Statue of Liberty during a trip to New York City. The height of the model i 9. 3 centimeter. Find the actual height x (in meter) of the Statue of Liberty
Answers
The actual height of the Statue of Liberty in meter is 93 m
What is proportionality?It is a constant relationship between different measurable quantities.
Data of the problem:
Scale 1:1000Height model = 9.3 cmWe calculate the proportion of the model scale:
1/1000 = 9.3/x
We clear the variable, noticing the rules of clearing:
What is multiplying goes to the other side of the equality dividing.What is dividing goes to the other side of the equality by multiplying.Clearing the variable we have:
1/1000 = 9.3/x
x* (1/1000) = 9.3
x = (9.3*1000) / 1
x = 9300 cm
Converting the centimeter to meters we have:
9300 cm * (1 m /100 cm) = 93 m
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the volume of a sphere is decreasing at a constant rate of 3 cubic centimeters per second. at the instant when the radius of the sphere is decreasing at a rate of 0.25 centimeter per second, what is the radius of the sphere?
Answers
As the volume of the sphere is decreasing at a constant rate of 3 cm³/s, the radius of the sphere is 0.98 cm
The volume of the sphere is given by:
V = 4/3. πr³
Where:
r = radius.
Take the derivative with respect to t
dV/dt = 4. πr² dr/dt
Data from the problem:
dV/dt = 3 cm³/s
dr/dt = 0.25 cm/s
Plug these parameters into the equation:
3 = 4. πr² (0.25)
πr² = 3
r² = 3/π
r = sqrt (3/π) = 0.98 cm
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Can you provide the solution for this exercise?
Let u(w) = −(b − w)c. What restrictions on w, b, and c are required to ensure that u(w) is strictly increasing and strictly concave? Show that under those restrictions, u(w) displays increasing absolute risk aversion.
Answers
under the restrictions that c is negative to ensure strict concavity, the utility function u(w) = -(b - w)c displays increasing absolute risk aversion.
To ensure that u(w) is strictly increasing, we need the derivative of u(w) with respect to w to be positive for all values of w. Taking the derivative, we have du(w)/dw = -c. For u(w) to be strictly increasing, -c must be positive, which implies c must be negative.
To ensure that u(w) is strictly concave, we need the second derivative of u(w) with respect to w to be negative for all values of w. Taking the second derivative, we have d²u(w)/dw² = 0. Since the second derivative is constant and negative, u(w) is strictly concave.
Now, let's examine the concept of increasing absolute risk aversion. If a utility function u(w) exhibits increasing absolute risk aversion, it means that as wealth (w) increases, the individual becomes more risk-averse.
In the given utility function u(w) = -(b - w)c, when c is negative (as required for strict concavity), the absolute risk aversion increases as wealth (w) increases. This is because the negative sign implies that the utility function is concave, indicating that the individual becomes more risk-averse as wealth increases.
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A restaurant offers a special pizza with any 5 toppings. if the restaurant has 16 topping from which to choose, how many different special pizzas are possible?
Answers
By permutation without repetition, there are 524160 possible different special pizzas.
The possible special pizzas can be calculated with permutation without repetition. The formula of permutation without repetition can be written as
P = n! / (n - k)!
where P is all of the possible combinations, n is the number of objects or elements, and k is how many numbers should be chosen.
From the question above, we know that :
n = 16
k = 5
By substituting the parameters, we can determine all of the possible different pizzas
P = n! / (n - k)!
P = 16! / (16 - 5)!
P = 16! / 11!
P = 16 x 15 x 14 x 13 x 12 x 11! / 11!
P = 524160
Hence, there are 524160 possible different special pizzas
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What is the approximate area of the circle shown below?
A. 94 cm2
B. 11,310 cm2
C. 2827 cm2
D. 188 cm2
Please help
Answers
Answer:
Step-by-step explanation:
we have the diameter =60 cm, the radius is 60/2=30 cm
A circle is 3.14*r^2=2827 cm^2
Answer: C. 2827 cm2
Step-by-step explanation:
The area of a circle is calculated by pir^2; r = the radius
In the diagram, the diameter is provided. The radius is half of the diameter. 60/2 = 30, so the radius is 30 cm.
Plug these values into the formula
A = 3.14*30^2
A = 2827.43, which would round to 2827 cm2
1, Suppose z varies directly with z and inversely with the square of y. If z = 12 when I = 3 and y what is z when 3 = 8 and y = 8? 2 = Question Help: D Video
Answers
The value of z when x = 8 and y = 9 is given as follows:
z = 32/27.
How to model the proportional relationship?A proportional relationship can be formed by one or multiple variables, and these variables are multiplied by the constant of proportionality k.
Suppose z varies directly with x and inversely with the square of y, hence the equation is given as follows:
z = kx/y².
z=18 when x=6 and y=2, hence the constant k is obtained as follows:
6k/4 = 18
6k = 72
k = 72/6
k = 12.
Hence the equation is:
z = 12x/y².
When x=8 and y=9, the value of z is given as follows:
z = 12 x 8/81
z = 96/81
z = 32/27.
Missing InformationThe problem is:
Suppose z varies directly with x and inversely with the square of y. If z=18 when x=6 and y=2, what is z when x=8 and y=9?
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The school library has 2,469 books. Two-thirds of them are paperbacks. How many
books are paperbacks?
Answers
Answer:
2469 x 2/3= 1646
personally, I solve it by dividing the 2469 by 3 to get 1/3.
2469/3= 823
then, double it to get 2/3.
823x2= 1646
see if you can find x ??
Answers
Answer:
just do 130 +5x+19=180
Answer:
Step-by-step explanation:
180° = 130°+5x+10
180 = 140+5X
-140 -140
40 = 5X
/5 /5
8 = X
Will give brainliest
Answers
Step-by-step explanation:
KM is a diagonal of a circle.
therefore, the arc angle KM is 180°.
the arc angle JKM is then the arc angle KM plus the arc angle JK.
arc angle JKM = 180 + 21 = 201°.
Answer:
arc JKM = 201°
Step-by-step explanation:
the measure of an arc is equal to the central angle it subtends , then
arc JK = 21°
given KM is a diameter of the circle then arc KM = 180°
then
arc JKM = JK + KM = 21° + 180° = 201°
is equal to 1,024. Evaluate the following expressions.
a. 46
b. 44
43.42
Answers
Answer:
What is equal to 1,024
Step-by-step explanation:
According to communication researchers, the ideal group size involves how many members?
A) 5 to 7 members
B) 15 to 17 members
C) 11 to 13 members
D) 3 to 4 members
E) 8 to 10 members
Answers
Ideal group size is 5 to 7 members, for work, social, and academic groups. Optimal interaction, decision-making, problem-solving, and logistics are possible, with reduced conflicts and power struggles.
The ideal group size is a topic that has been widely studied by communication researchers. While there is no universally agreed-upon answer, many researchers suggest that a group size of 5 to 7 members is optimal for a range of different types of groups, including work teams, social groups, and academic groups. One reason why this group size is considered ideal is that it allows for optimal interaction and participation. In small groups, each member has a greater opportunity to speak and be heard, and there is less likelihood of individuals being drowned out or overlooked. This can lead to more productive and satisfying group interactions, as well as increased engagement and motivation among group members.
Another reason why a group size of 5 to 7 members is preferred is that it allows for effective decision-making and problem-solving. In larger groups, it can be difficult to achieve consensus or to reach a decision that reflects the needs and perspectives of all members. Conversely, groups that are too small may lack diversity of thought and expertise, which can limit the range of possible solutions or approaches to a problem.
In addition to these benefits, a group size of 5 to 7 members may also be more manageable in terms of logistics and group dynamics. For example, it may be easier to schedule meetings and coordinate group activities with a smaller group, and there may be less potential for conflicts or power struggles to arise among members.
It's worth noting that while a group size of 5 to 7 members is often recommended, there are certainly situations in which larger or smaller groups may be appropriate or necessary. For example, certain types of projects or initiatives may require a larger pool of resources or expertise, while others may benefit from a more intimate and tightly-knit group dynamic. Nonetheless, the research suggests that a group size of 5 to 7 members is a good starting point for most types of groups.
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the domain of consists of numbers x for which not equals equals not equals 0 that are in the domains of both f and g. both f and g. either f or g.
Answers
The statement is true that the domain of f(x)/g(x) consists of numbers "x" that are in the domains of both f(x) and g(x) for which g(x) is not equal to 0.
A function's area is the set of its available inputs, or the set of enter values for which the characteristic is described. The area is the collection of objects that the system will take delivery of as inputs within the characteristic machine paradigm.For the function f(x)/g(x) to be defined, g(x) must now not be equal to 0.hence, the domain of the characteristic f(x)/g(x) consists of all of the numbers "x" which can be inside the domain names of each the functions f(x) and g(x) for which g(x) is not 0.To learn more about functions, visit :
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the sum of the digits of a two-digit number is 8 . when 36 is subtracted from this number, its digits are reversed. find the original number.0
Answers
The original number is 80 and the sum of the digits of a two-digit number is 8 . When 36 is subtracted from this number, its digits are reversed.
Let us suppose that the two-digit number is '10x + y' (as the number is greater than 9 and less than 100, we can represent it as '10x + y' where 'x' is tens digit and 'y' is units digit).
According to the question,
The sum of the digits of a two-digit number is 8
⇒ x + y = 8
Also,
When 36 is subtracted from this number, its digits are reversed
⇒ 10x + y - 36 = 10y + x + 36
⇒ 9x - 9y = 72
⇒ x - y = 8
Therefore, the equations are:
x + y = 8 ........ (1)
x - y = 8 ........ (2)
By adding equations (1) and (2), we get:
2x = 16
⇒ x = 8
By substituting value of 'x' in equation (1), we get:
y = 0
Therefore, the original number is 80.
Conclusion: So, the original number is 80 and the sum of the digits of a two-digit number is 8 . When 36 is subtracted from this number, its digits are reversed.
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The sum of the digits of a two-digit number is 8 . when 36 is subtracted from this number, its digits are reversed.
The original number is 62.
Let's assume the two-digit number is represented by 10a + b, where a and b are the digits of the number.
According to the given information, we have two conditions:
1. The sum of the digits is 8: a + b = 8.
2. When 36 is subtracted from the number, its digits are reversed: (10a + b) - 36 = 10b + a.
Now we can solve these equations to find the values of a and b.
From equation 1, we have a + b = 8.
From equation 2, we have 10a + b - 36 = 10b + a.
Simplifying equation 2, we get:
9a - 9b = 36.
Dividing both sides by 9, we have:
a - b = 4.
Now we have a system of equations:
a + b = 8,
a - b = 4.
Adding these two equations together, we get:
2a = 12,
a = 6.
Substituting the value of a into one of the equations, we can find b:
6 + b = 8,
b = 2.
Therefore, the original number is 62.
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someone please help me
Answers
9514 1404 393
Answer:
C. f(x) = 2(3^x)
Step-by-step explanation:
The y-intercept is the function value when x=0. Your choices are ...
A. f(0) = 1 +2 = 3
B. f(0) = 3(1) = 3
C. f(0) = 2(1) = 2 . . . . . the function of interest
D. f(0) = 2(1) -2 = 0
Find the volume of this object.Use 3 for π.Volume of a CylinderV= πr²h4 ft9 ft5 ft5 ft5 ftVolume of aRectangular PrismV = whV ≈ [?]ft³Enter
Answers
Mathematics → Solid Figures → Volume
The volume of a cylinder is:
\(V=\pi r^2h\)In this question,
r = 2 ft (4/2 =2)
h = 5 ft
Then, the volume of the cylinder is (Vc):
\(\begin{gathered} Vc=\pi *2^2*5 \\ Vc=\pi *4*5 \\ Vc=20\pi \end{gathered}\)Using π = 3:
\(\begin{gathered} Vc=3*20 \\ Vc=60ft^3 \end{gathered}\)The volume of a rectangular prism (Vr) is:
\(Vr=lwh\)In this question:
l = 9 ft
w = 5 ft
h = 5 ft
Then, the volume of the rectangular prism is:
\(\begin{gathered} V=9*5*5 \\ V=225ft^3 \end{gathered}\)The volume of the object (V) is the sum of the volumes:
\(\begin{gathered} V=Vc+Vr \\ V=60+225 \\ V=285ft^3 \end{gathered}\)Answer: 285 ft³.
Simplify the following :
\(\begin{gathered} \: \boxed{\sf \: {\bigg( {x}^{ \frac{b}{b - c} } \bigg) }^{ \frac{1}{b - a} } \times {\bigg( {x}^{ \frac{c}{c - a} } \bigg) }^{ \frac{1}{c - b} } \times {\bigg( {x}^{ \frac{a}{a - b} } \bigg) }^{ \frac{1}{a - c} }} \end{gathered}\)
Answers
Answer:
1
Step-by-step explanation:
\(\left(x^{\tfrac b{b-c}} \right)^{\tfrac 1{b-a}} \times \left(x^{\tfrac c{c-a}} \right)^{\tfrac 1{c-b}} \times \left(x^{\tfrac a{a-b}} \right)^{\tfrac 1{a-c}}\\\\\\=x^{\tfrac{b}{(b-c)(b-a)}} \cdot x^{\tfrac{c}{(c-a)(c-b)}} \cdot x^{\tfrac{a}{(a-b)(a-c)}}~~~~~;\left[\text{Apply exponent rule:}~ (a^m)^n = a^{mn} \right]\\\\=x^{\tfrac{b}{(b-c)(b-a)} + \tfrac{c}{(c-a)(c-b)} + \tfrac{a}{(a-b)(a-c)}} ~~~~~~~~;\left[\text{Apply exponent rule:}~ a^m \cdot a^n = a^{m+n} \right]\\\\\)
\(=x^{\tfrac{b}{-(c-b)(b-a)} + \tfrac{c}{-(a-c)(c-b)} + \tfrac{a}{-(b-a)(a-c)}} \\\\=x^{-\tfrac{b}{(c-b)(b-a)} - \tfrac{c}{(a-c)(c-b)} -\tfrac{a}{(b-a)(a-c)}} \\\\=x^{-\left[\tfrac{b(a-c)+c(b-a)+a(c-b)}{(c-b)(b-a)(a-c)} \right]}\\\\=x^{-\left[\tfrac{ba-bc+cb-ac+ac-ab}{(c-b)(b-a)(a-c)} \right]}\\\\=x^{\tfrac{-ba+bc-cb+ac-ac+ab}{(c-b)(b-a)(a-c)} }\\\\=x^{\tfrac{0}{(c-b)(b-a)(a-c)}}\\\\=x^0\\\\=1\)