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
What is a pyramid?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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A 64-ounce bottle of
orange juice hos 48 ounces of water, which juice has a greater percent of water
What percent of the bottle of apple juice is water
% water
% water
40
Answer:
Orange juice bottle has greater percent of water.
Percent of water in apple juice bottle=40%
Step-by-step explanation:
We are given that
Total mass of apple juice bottle=80 ounce
Apple juice contains water=32 ounces
Total mass of orange juice bottle=64 ounce
Orange juice bottle contain water=48 ounces
We have to find that which juice has greater percent of water and find the percent of water in the bottle of apple juice .
Percent of water in apple juice=[tex]\frac{water}{total\;mass}\times 100=\frac{32}{80}\times 100=[/tex]40%
Percent of water in orange juice bottle=[tex]\frac{48}{64}\times 100=[/tex]75%
Orange juice bottle has greater percent of water.
Step 1: -3(+ 2) = 5(x - 7)
Step 2: -3.1 - 6 = 50 – 35
Step 3: _88 – 6 = -35
Step 4: -80 = -29
Step 5: I = 22
Between which two steps did Henry apply the distributive property?
steps 1 and 2
steps 2 and 3
steps 3 and 4
steps 4 and 5
Answer:
The answer is Steps 1 and 2. Step 1: -3(x+2) = 5(x-7)
Step 2: -3x-6 = 5x-35
Step-by-step explanation:
Ron wants to calculate the sales tax on two items. He is purchasing a helmet for $42 and gloves for $5.65. Sales tax is 7%. Which expression shows how Ron should calculate his total sales tax?
A
$42 + $5.65 x 7
B
$42 + $5.65 x 0.7
C
($42 + $5.65) x 0.7
D
($42 + $5.65) x 0.07
In ΔWXY, the measure of ∠Y=90°, XW = 25, YX = 24, and WY = 7. What is the value of the cosine of ∠W to the nearest hundredth?
Answer:
7/25
Step-by-step explanation:
Just did on delta math
Monique ran in a 5-kilometer rece. How many meters did Monique run?
Answer:5000
Step-by-step explanation:
A kilometer is equal to 1000 meters.
5*1000=5000.
Answer:
5000 meters
Step-by-step explanation:
It is given that Monique ran in a 5 kilometer race.
Now suppose that out of the 10 dishes that the restaurant offers, only 3 of them are vegetarian. If Fiona must select a vegetarian option on Friday, how many ways are there for her to select her lunches?
Answer:
The number of ways she can select her lunches N ;
N = 10×10×10×10×3 = 30000 ways
Step-by-step explanation:
Number of week days = 5
Total number of dishes = 10
Number of vegetarian dishes = 3
Note; on other week days apart from Friday she can select any lunch she want vegetarian or not...
Therefore, on Monday to Thursday she has ten choices per day, and on Friday she has 3 choices
The number of ways she can select her lunches N ;
N = 10×10×10×10×3 = 30000 ways
According to a Pew Research Center, in May 2011, 35% of all American adults had a smart phone (one which the user can use to read email and surf the Internet). A communications professor at a university believes this percentage is higher among community college students.
She selects 300 community college students at random and finds that 120 of them have a smart phone. In testing the hypotheses: H0: p = 0.35 versus Ha: p > 0.35, she calculates the test statistic as Z = 1.82.
Use the Normal Table to help answer the p-value part of this question.
Click here to access the normal table.
1. There is enough evidence to show that more than 35% of community college students own a smart phone (P-value = 0.034).
2. There is enough evidence to show that more than 35% of community college students own a smart phone (P-value = 0.068).
3. There is not enough evidence to show that more than 35% of community college students own a smart phone (P-value = 0.966).
4. There is not enough evidence to show that more than 35% of community college students own a smart phone (P-value = 0.034).
Answer:
[tex]p_v =P(z>1.82)=0.034[/tex]
Assuming a standard significance level of [tex]\alpha=0.05[/tex] the best conclusion for this case would be:
4. There is not enough evidence to show that more than 35% of community college students own a smart phone (P-value = 0.034).
Because [tex] p_v <\alpha[/tex]
If we select a significance level lower than 0.034 then the conclusion would change.
Step-by-step explanation:
Data given
n=300 represent the random sample taken
X=120 represent the people who have a smart phone
[tex]\hat p=\frac{120}{300}=0.4[/tex] estimated proportion of people who have a smart phone
[tex]p_o=0.35[/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)
System of hypothesis
We need to conduct a hypothesis in order to test the claim that the true proportion of people who have a smart phone is higher than 0.35, the system of hypothesis are.:
Null hypothesis:[tex]p\leq 0.35[/tex]
Alternative hypothesis:[tex]p > 0.35[/tex]
The statistic is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.4 -0.35}{\sqrt{\frac{0.35(1-0.35)}{300}}}=1.82[/tex]
Statistical decision
Since is a right tailed test the p value would be:
[tex]p_v =P(z>1.82)=0.034[/tex]
Assuming a standard significance level of [tex]\alpha=0.05[/tex] the best conclusion for this case would be:
4. There is not enough evidence to show that more than 35% of community college students own a smart phone (P-value = 0.034).
Because [tex] p_v <\alpha[/tex]
If we select a significance level lower than 0.034 then the conclusion would change.
Gender in the Population of Part-time College Students According to a 2010 report from the American Council on Education, females make up 57% of the U.S. college population. With the rising costs of education and a poor economy, many students are working more and attending college part time. We anticipate that if we look at the population of part-time college students, a larger percentage will be female. Let’s say we predict that 60% of part-time college students are female. We don’t have information about the population of part-time college students, so we select a random sample of 25 part-time college students and calculate the proportion of the sample that is female. We don’t expect the sample proportion to be exactly 0.60. So, how much could the sample proportion vary from 0.60 for us to feel confident in our prediction?To answer this question, we need to understand how much sample proportions will vary if the parameter is 0.60.Refer to the previous example for the following questions. These questions focus on how the proportion of females will vary in random samples if we assume that 0.60 of the population of part-time college students is female.
Answer:
These questions focus on how the proportion of females will vary in random samples if we assume that 0.60 of the population of part-time college students is female.
1. Before we use a simulation to simulate the selection of random samples from this population, let’s make sure we are clear about who is in the population. For this situation which statement best describes the population? part-time college students
2. What are we assuming to be true about the population? The proportion of the population that is female is 0.60
3. Which of the following sequences of sample proportions is the most likely to occur in random samples of 25 students from this population? 0.56, 0.60, 0.44, 0.68, 0.76
4)
I
You deposit $2500 in an account that pays 6 percent annual interest. Find the balance after 3
years if the interest is compounded with the given frequency.
Answer:
$2,977.54
Step-by-step explanation:
You are going to use the compound interest formula:
[tex]A=P(1+\frac{r}{n} )^{nt}[/tex]
P = initial balance
r = interest rate
n = number of times compounded annually
t = time
First, change 6% into the decimal form:
6% -> [tex]\frac{6}{100}[/tex] -> 0.06
Next, lets plug in the values:
[tex]A=2,500(1+\frac{0.06}{1} )^{3(1)}[/tex]
[tex]A=2,977.54[/tex]
Your answer will be $2,977.54
There are 6 cars traveling together. Each car has two people in front and three people in back. Explain how to use the situation to illustrate the distributive property.
The situation can illustrate the distributive property in math (6*(2+3)=6*2 + 6*3), representing 'a' as six cars, and 'b' and 'c' as the people in each car.
Explanation:The situation described can be used to illustrate the distributive property in mathematics. The distributive property states a(b + c) = ab + ac. In this scenario, the six cars can be represented as 'a', while the 'b' and 'c' can represent the two people in front and the three in the back. Therefore, the total number of people in all cars is 6*(2+3) which according to the distributive property equals 6*2 + 6*3. That is, 12 people in the front seats and 18 in the back seats, summed up to be 30 people in total.
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Final answer:
The distributive property states that the product of a number and a sum is equal to the sum of the individual products. In this situation with the cars, we can use the distributive property to calculate the total number of people in all the cars.
Explanation:
The distributive property in mathematics states that the product of a number and a sum is equal to the sum of the individual products. In this situation with the cars, we can use the distributive property to calculate the total number of people in all the cars. Each car has 2 people in front and 3 people in back, so we can write the expression as (2 + 3) * 6. Using the distributive property, we can simplify this expression to 2 * 6 + 3 * 6, which equals 12 + 18, or 30 people in all the cars.
An e-commerce research company claims that 60% or more graduate students have bought merchandise on-line at their site. A consumer group is suspicious of the claim and thinks that the proportion is lower than 60%. A random sample of 80 graduate students show that only 44 students have ever done so. Is there enough evidence to show that the true proportion is lower than 60%? Assume that significance level of 0.05. Use the hypothesis testing template provided.'
Answer:
We accept H₀ we don´t have enough evidence to conclude that a consumer group position is correct
Step-by-step explanation:
We have a case of test of proportion, as a consumer group is suspicious of the claim and think the proportion is lower we must develop a one tail test (left tail) Then
1.- Test hypothesis:
Null hypothesis H₀ P = P₀
Alternative hypothesis Hₐ P < P₀
2.- At significance level of α = 0,05 Critical value
z(c) = -1,64
3.-We compute z(s) value as:
z(s) = ( P - P₀ )/ √P*Q/n where P = 44/80 P = 0,55 and Q = 0,45
P₀ = 0,6 and n = 80
Plugging all these values in the equation we get:
z(s) = ( 0,55 - 0,6 ) / √(0,2475/80)
z(s) = - 0,05/ √0,0031
z(s) = - 0,05/0,056
z(s) = - 0,8928
4.-We compare z(s) and z(c)
z(s) > z(c) -0,8928 on the left side it means that z(s) is in the acceptance region so we accept H₀
You go to Home Depot to buy some materials for a project. Your total bill for materials only was $45.90. You go to the check out where you get charged taxes of 7.2%. That gets added to your bill. You then have a coupon for $15 off your total purchase after taxes. What is the amount you actually pay for your bill after taxes and coupons are applied?
Answer:
$34.20
Step-by-step explanation:
First, find the cost with tax
You are paying 7.2% tax, but you are also paying for 100% of the bill
100%+7.2%=107.2%
So, you are really paying for 107.2% of the bill
Convert 107.2% to a decimal by dividing by 100 or moving the decimal 2 spaces to the right
107.2/100=1.072
Multiply the decimal by the cost of the bill
1.072*45.90=49.2048
But, we also have a coupon for $15 off
Subtract 15 from the cost with tax
49.2048-15=34.2048
Round to the nearest cent/hundreth
$34.20
Hope this helps! :)
The diagram shows the distance
find the area of the polygons 5 cm 5 cm 8 cm square centimeters
Answer:
Answer: 600
Step-by-step explanation:
5x5=25
3x8=24
25x24=600
FInd the measure of RST.
At the Canada Open Tennis Championship, a statistician keeps track of every serve that a player hits during the tournament. The statistician reported that the mean serve speed of a particular player was 99 miles per hour (mph) and the standard deviation of the serve speeds was 15 mph.
If nothing is known about the shape of the distribution, give an interval that will contain the speeds of at least eight-ninths of the player's serves.
a) 54 mph to 144 mph
b) 39 mph to 159 mph
c) 144 mph to 189 mph
d) 69 mph to 129 mph
Answer:
a) 54 mph to 144 mph
Step-by-step explanation:
We don't know the shape of the distribution, so we use Chebyshev's Theorem to solve this question. It states that:
At least 75% of the measures are within 2 standard deviations of the mean.
At least 89% of the measures are within 3 standard deviations of the mean.
At least eight-ninths of the player's serves.
8/9 is approximately 89%
So
Mean: 99, standard deviation: 15
99 - 3*15 = 54
99 + 3*15 = 144
So the correct answer is:
a) 54 mph to 144 mph
the joint frequency for females is
Answer:
55%
Step-by-step explanation:
Coralee invests $5,000 in an account that compounds interest monthly and earns 7%. How long will it take her money to double?
Answer:
It will take 10 years for her money to double.
Step-by-step explanation:
The compound interest formula is given by:
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
Where A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per year and t is the number of years the money is invested or borrowed for.
In this exercise:
We want to find t for which the money doubles, that is, t when A = 2P.
Compounded monthly, an year has 12 months, so n = 12
Interest rate of 7%, so r = 0.07.
The following logarithm property is used:
[tex]\log{a^{t}} = t\log{a}[/tex]
So
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
[tex]2P = P(1 + \frac{0.07}{12})^{12t}[/tex]
[tex](1.0058)^{12t} = 2[/tex]
[tex]\log{(1.0058)^{12t}} = \log{2}[/tex]
[tex]12t\log{1.0058} = \log{2}[/tex]
[tex]t = \frac{\log{2}}{12\log{1.0058}}[/tex]
[tex]t = 10[/tex]
It will take 10 years for her money to double.
HI PLEASE HELP ME WITH MY CALCULUS 1 HW? I AM REALLY STUCK. I need help with parts d,e,g.
(d) The particle moves in the positive direction when its velocity has a positive sign. You know the particle is at rest when [tex]t=0[/tex] and [tex]t=3[/tex], and because the velocity function is continuous, you need only check the sign of [tex]v(t)[/tex] for values on the intervals (0, 3) and (3, 6).
We have, for instance [tex]v(1)\approx-0.91<0[/tex] and [tex]v(4)\approx0.91>0[/tex], which means the particle is moving the positive direction for [tex]3<t<6[/tex], or the interval (3, 6).
(e) The total distance traveled is obtained by integrating the absolute value of the velocity function over the given interval:
[tex]\displaystyle\int_0^6|v(t)|\,\mathrm dt=\int_0^3-v(t)\,\mathrm dt+\int_3^6v(t)\,\mathrm dt[/tex]
which follows from the definition of absolute value. In particular, if [tex]x[/tex] is negative, then [tex]|x|=-x[/tex].
The total distance traveled is then 4 ft.
(g) Acceleration is the rate of change of velocity, so [tex]a(t)[/tex] is the derivative of [tex]v(t)[/tex]:
[tex]a(t)=v'(t)=-\dfrac{\pi^2}9\cos\left(\dfrac{\pi t}3\right)[/tex]
Compute the acceleration at [tex]t=4[/tex] seconds:
[tex]a(t)=\dfrac{\pi^2}{18}\dfrac{\rm ft}{\mathrm s^2}[/tex]
(In case you need to know, for part (i), the particle is speeding up when the acceleration is positive. So this is done the same way as part (d).)
Steve wants to know what the average ACT scores of students graduated from his high school is. He randomly selected 25 students from an email list and asked them to provide the information. Of them 20 students replied and the summary of the ACT score is as below: Sample size Mean STD Min Max 20 21.5 5.7 15 26 (1) What is the sample of the study and what is the population of the study? (2) Steve is told that the population standard deviation for ACT score is 4.6 for his high school, estimate the population average ACT score of his high school graduates with 86% confidence level. (3) If the population standard deviation is not provided, estimate the population average ACT score of his high school graduates with 95% confidence level.
Answer:
[a]. sample is 25 randomly selected students
also population is all students
[b]. (20.0, 23.0)
[c]. (8.8, 24.2)
Step-by-step explanation:
here is a step by step process to solving this, i hope you find this helpful.
1). The population is all students who graduated from his school while the
sample is 25 randomly selected students from the email list
2). given alpha = 1-0.86=0.14
critical z value(two tailed) for 86% confidence level is:
z=normsinv(0.07) or normsinv(0.93)=1.476
Margin of error=1.476*(4.6/SQRT(20))=1.5
86% confidence interval for population mean
=21.5+/-1.5
=(20.0, 23.0)
3). if population standard deviation is not known,we will use t distribution.
df=20-1=19
t*=tinv(0.05,19)=2.093
Margin of error=2.093*(5.7/sqrt(20))=2.7
95% confidence interval for population mean
=21.5+/-2.7
=(18.8, 24.2)
cheers i hope this helps!!!!
(1 point) A computer retail store has 8 personal computers in stock. A buyer wants to purchase 4 of them. Unknown to either the retail store or the buyer, 4 of the computers in stock have defective hard drives. Assume that the computers are selected at random. (a) In how many different ways can the 4 computers be chosen
Answer:
Total no of different ways can the 4 computers be chosen = 70
Step-by-step explanation:
Given -
A computer retail store has 8 personal computers in stock.A buyer wants to purchase 4 of them.
(Using the combination formula)
The combination of n events taking r at a time = [tex]\mathbf{\binom{n}{r}}[/tex]
where n = 8 and r = 4
Total no of different ways can the 4 computers be chosen = [tex]\mathbf{\binom{8}{4}}[/tex]
= [tex]\boldsymbol{\frac{8!}{(4!)(4!)}}[/tex]
= 70
The number of different ways that 4 computers can be chosen from a stock of 8 is 70.
Explanation:The number of different ways that 4 computers can be chosen from a stock of 8 is determined by the concept of combinations. In this case, we want to find the number of combinations of 8 items taken 4 at a time. The formula for combinations is given by:
C(n, r) = n! / r!(n-r)!
where n is the total number of items and r is the number of items to be chosen. Applying the formula to this problem, we have:
C(8, 4) = 8! / 4!(8-4)!
Simplifying the expression, we get:
C(8, 4) = 8! / 4!4!
Using the factorial function, we can compute:
C(8, 4) = 8 × 7 × 6 × 5 / (4 × 3 × 2 × 1)
C(8, 4) = 70
Therefore, there are 70 different ways that the 4 computers can be chosen from the stock of 8.
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which point represent on the number line -3/2
Answer:
Half way in between negative 1 and negative 2.
Step-by-step explanation:
HELP ASAP PLEASE!!!
Which of the following angles are shown in the drawing?
a. ADB
b. D
c. CBD
d. 1
Answer:
Adb , cbd, 1
Step-by-step explanation:
Answer:
ADB CBD 1
Step-by-step explanation:
Find the volume of the sphere.
Either enter an exact answer in terms of pi or use 3.14 and round to the nearest hundredth
Answer:
The volume is 2144.66
Step-by-step explanation:
The equation i used when solving this question.
V=4/3πr3
The volume of sphere when rounded to nearest 100th will be 2143.574 cubic units
Given: a sphere of radius 8 units
To find: Volume of sphere
The volume of a sphere can be calculated using the formula [tex]V = \frac{4}{3}\pi r^3[/tex], where r is the radius of the sphere. To find the volume, plug in the value of the radius into the formula and calculate the result.
[tex]V = \frac{4}{3}\pi r^3\\[/tex]
[tex]V = \frac{4}{3}\pi (8)^3\\[/tex]
[tex]V = \frac{4}{3}\pi (8)^3 = 682.667 \pi[/tex]
If we insert the value of [tex]\pi = 3.14[/tex] we get:
[tex]V = 682.667 \times 3.14 = 2143.57438[/tex]
When rounded to nearest 100th we will get volume equal to 2143.574 cubic units.
A diver dives from a cliff with a height of 144 feet. His height, h, in feet is given by the equation h=-16t^2+144 , where t is the time in seconds the diver has fallen. How many seconds will it take for the diver to reach the water?
Answer:
It takes 3 seconds for the diver to reach the water.
Step-by-step explanation:
The height, in feet, is given by the following equation:
[tex]h(t) = -16t^{2} + 144[/tex]
How many seconds will it take for the diver to reach the water?
He reaches the water when his height is 0, that is, when h(t) = 0. So
[tex]h(t) = -16t^{2} + 144[/tex]
[tex]0 = -16t^{2} + 144[/tex]
[tex]16t^{2} = 144[/tex]
[tex]t^{2} = \frac{144}{16}[/tex]
[tex]t^{2} = 9[/tex]
[tex]t = \pm \sqrt{9}[/tex]
[tex]t = \pm 3[/tex]
There are no negative instant of time, so just t = 3.
It takes 3 seconds for the diver to reach the water.
Last month, Ginger's hotel ran an occupancy rate of 85%. Her competitive set had 150,000 room nights included within it that were available for sale during that month. During that month, the competitive set sold a total of 115,000 rooms. What was the approximate occupancy rate INDEX last month for Ginger's hotel? It's one of the following answer; 78%, 92%, 87% or 39%? Provide details.
Given Information:
Occupancy rate of Ginger’s hotel = 85%
Competitive set available rooms = 150,000
Competitive set sold rooms = 115,000
Required Information:
Occupancy rate INDEX = ?
Answer:
Occupancy rate INDEX = 111%
Step-by-step explanation:
The occupancy rate INDEX is given by
Occupancy rate INDEX = Occupancy rate of Ginger’s hotel/Occupancy rate of competitive set
We are given the occupancy rate of Ginger’s hotel so we need to first find out occupancy rate of competitive set which is given by
Occupancy rate of competitive set = Competitive set sold rooms/Competitive set available rooms
Occupancy rate of competitive set = 115,000/150,000
Occupancy rate of competitive set = 0.7667
Occupancy rate of competitive set ≈ 76.67%
Finally, we can now find the occupancy rate INDEX
Occupancy rate INDEX = Occupancy rate of Ginger’s hotel/Occupancy rate of competitive set
Occupancy rate INDEX = 85/76.67
Occupancy rate INDEX = 1.108
Occupancy rate INDEX = 110.8%
Occupancy rate INDEX ≈ 111%
Therefore, the approximate occupancy rate INDEX last month for Ginger's hotel is 111%
An infinite geometric series converges if the common ratio is
An infinite geometric series would only converge if the common ratio is a proper fraction.
What is an infinite geometric series?An infinite geometric series can be defined as the sum of a geometric sequence that typically has a constant common ratio between successive terms but no last term.
In Mathematics, it is a fact that if the common ratio of an infinite geometric series is a proper fraction, this would make it to converge because each successive term gets smaller and smaller.
In conclusion, an infinite geometric series would only converge if the common ratio is a proper fraction such as 3/6.
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Answer:
c. a proper fraction
b and d. -2/3 and 3/4
c. 80/3
Step-by-step explanation:
...
Explaining What Causes Seasons
Which factors cause Earth to experience seasons? Check all that apply.
the speed of Earth’s rotation
the tilt of Earth’s axis
the directness of the Sun’s rays
the distance from the Sun
the distance from the equator
the altitude of an area
Answer:
The factors that cause Earth to experience seasons are:
1. the tilt of Earth’s axis
2. the directness of the Sun’s rays
The factors that cause Earth to experience seasons are:
1. the tilt of Earth’s axis
2. the directness of the Sun’s rays
Factors that result in the earth experiencing the seasons should be:The reason why the Earth contains various seasons because it deals with the variation of the sun's rays angles and the earth titles to the 23.5 degrees on its axis. Also along with the earth rotation, the earth orbits should be around to the sun because of which various parts should be exposed to the different amount of lights
Therefore, the above two points should be considered.
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Steve likes to entertain friends at parties with "wire tricks." Suppose he takes a piece of wire 52 inches long and cuts it into two pieces. Steve takes the first piece of wire and bends it into the shape of a perfect circle. He then proceeds to bend the second piece of wire into the shape of a perfect square. What should the lengths of the wires be so that the total area of the circle and square combined is as small as possible
Answer:
Steve should use:
8.53 Inch of Wire to make the circle43.47 Inches of Wire to make the Square.Step-by-step explanation:
Let Steve cut the wire so that the first piece has length x.
Therefore, the second piece will have a length of (52 - x).
The wire of length x is used to make a circle
Circumference of a Circle,
[tex]c = 2\pi r\\Therefore\\2\pi r=x\\r=\dfrac{x}{2\pi}[/tex]
[tex]\text{Area of a circle, A}=\pi r^2= \pi (\dfrac{x}{2\pi})^2=\pi (\dfrac{x^2}{4\pi^2})\\A=\dfrac{x^2}{4\pi}[/tex]
The wire of length (52-x) is used to make a square.
[tex]\text{Side length of the Square,}[/tex] [tex]s= \dfrac{52-x}{4}[/tex]
[tex]\text{Area of the Square},s^2= \left(\dfrac{52-x}{4}\right)^2=\dfrac{(52-x)^2}{16}=\dfrac{x^2-104x+2704}{16}[/tex]
Total Area = Area of Circle + Area of Square
[tex]Area=\dfrac{x^2}{4\pi}+\dfrac{x^2-104x+2704}{16}[/tex]
Let us simplify the expression
[tex]Area=\dfrac{16x^2+\pi(x^2-104x+2704)}{16\pi}\\=\dfrac{16x^2+\pi x^2-104\pix+2704\pi}{16\pi}\\=\dfrac{x^2(16+\pi)-104\pi x+2704\pi}{16\pi}\\=\dfrac{x^2(16+\pi)}{16\pi}-\dfrac{104\pi x}{16\pi}+\dfrac{2704\pi}{16\pi}\\A=\dfrac{x^2(16+\pi)}{16\pi}-6.5x+169[/tex]
This is the function of a parabola which opens up.
To find where A is minimum, find the axis of symmetry.
[tex]$Using \: x=-\frac{b}{2a}[/tex]
[tex]a=\dfrac{16+\pi}{16\pi}, b=-6.5\\ x=-\dfrac{-6.5}{2(\dfrac{16+\pi}{16\pi})}=8.53\:Inches[/tex]
Steve should cut the wire so that the length of wire used to make a circle is 8.53 Inches.
Length of wire used to make the square =52-8.53=43.47 Inches
An engineer designed a valve that will regulate water pressure on an automobile engine. The engineer designed the valve such that it would produce a mean pressure of 7.4 pounds/square inch. It is believed that the valve performs above the specifications. The valve was tested on 130 engines and the mean pressure was 7.6 pounds/square inch. Assume the standard deviation is known to be 0.9. A level of significance of 0.02 will be used. Make a decision to reject or fail to reject the null hypothesis. Make a decision.
Answer:
[tex]z=\frac{7.4-7.6}{\frac{0.9}{\sqrt{130}}}=-2.534[/tex]
[tex]z_{\alpha}=-2.054[/tex]
If the calculated value is less than the critical value we reject the null hypothesis.
P value
The p value for this test would be:
[tex]p_v =P(Z<-2.534)=0.0056[/tex]
Since the p value is lower than the significance level given we have enough evidence to reject the null hypothesis at the 25 of significance level given.
Step-by-step explanation:
Information given
[tex]\bar X=7.6[/tex] represent the sample mean
[tex]\sigma=0.9[/tex] represent the population deviation
[tex]n=130[/tex] sample size
[tex]\mu_o =7.4[/tex] represent the value that we want to check
[tex]\alpha=0.02[/tex] represent the significance level for the hypothesis test.
z would represent the statistic
[tex]p_v[/tex] represent the p value for the test
System of hypothesis
We want to check if the mean pressure is less then 7.6 pounds/square inch, the system of hypothesis are:
Null hypothesis:[tex]\mu \geq 7.6[/tex]
Alternative hypothesis:[tex]\mu < 7.6[/tex]
The statistic would be:
[tex]z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}[/tex] (1)
Replacing the values we got:
[tex]z=\frac{7.4-7.6}{\frac{0.9}{\sqrt{130}}}=-2.534[/tex]
Critical value
we need to find a critical value who accumulates 0.02 of the area in the left of the normal standard distribution and we got:
[tex]z_{\alpha}=-2.054[/tex]
If the calculated value is less than the critical value we reject the null hypothesis.
P value
The p value for this test would be:
[tex]p_v =P(Z<-2.534)=0.0056[/tex]
Since the p value is lower than the significance level given we have enough evidence to reject the null hypothesis at the 25 of significance level given.