A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.3 ft/s, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 8 ft from the wall? (That is, find the angle's rate of change when the bottom of the ladder is 8 ft from the wall.)
Answers
so, when r = 10 and x = 8, the vertical distance is 6, namely in pythagorean theorem lingo, b = 6.
let's keep in mind that, the ladder is not growing any longer or shrinking, and therefore is constantly always just 10, that matters, since is just a scalar value and also because the derivative of a constant is 0.
[tex]\bf cos(\theta )=\cfrac{x}{r}\implies cos(\theta )=\cfrac{1}{10}x\implies \stackrel{chain~rule}{-sin(\theta )\cfrac{d\theta }{dt}}=\cfrac{1}{10}\cdot\stackrel{chain~rule}{\cfrac{dx}{dt}\cdot 1}[/tex]
[tex]\bf -sin(\theta )\cfrac{d\theta }{dt}=-\cfrac{1}{10}\cdot \cfrac{dx}{dt}\implies \cfrac{d\theta }{dt}=-\cfrac{1}{-10sin(\theta )}\cdot \cfrac{dx}{dt} \\\\\\ \begin{cases} \frac{dx}{dt}=1.3\\ sin(\theta )=\frac{6}{10} \end{cases}\implies \cfrac{d\theta }{dt}=-\cfrac{1}{10\cdot \frac{6}{10}}\cdot 1.3\implies \cfrac{d\theta }{dt}=-\cfrac{1.3}{6}~radians[/tex]
The angle's rate of change when the bottom of the ladder is 8ft from the wall is
[tex]\dfrac{-1.3}{6} rad/s[/tex]
It is given the Length of Ladder [tex](h)[/tex] is [tex]10ft.[/tex]. and the distance between the bottom of the ladder to the wall [tex](r)[/tex] is [tex]8ft.[/tex] as shown in the below figure.
By using the Pythagoras Theorem
[tex]b=\sqrt{10^{2}-8^{2} }\\=\sqrt{36} \\=6ft.[/tex]
and
[tex]cos(\theta)= r/h\\cos(\theta)=r/10[/tex]
Differentiating both sides with respect to [tex]'t'[/tex] by using the chain rule
[tex]-sin(\theta)\dfrac{\mathrm{d}\theta }{\mathrm{d} t}=\dfrac{1}{10} \dfrac{\mathrm{d}r }{\mathrm{d} t}\\\\\dfrac{\mathrm{d}\theta }{\mathrm{d} t}=\dfrac{1}{10} \dfrac{\mathrm{d}r }{\mathrm{d} t} \dfrac{1}{-sin(\theta)} } ......(eq. 1)[/tex]
given
[tex]\dfrac{\mathrm{d}r }{\mathrm{d} t}=1.3ft./s\\sin(\theta)=\frac{6}{10}[/tex]
putting this in eq.1, we get
[tex]\dfrac{\mathrm{d} \theta}{\mathrm{d} t} = \dfrac{1.3}{10} (\dfrac{1}{-\frac{6}{10} }) \\\dfrac{\mathrm{d} \theta}{\mathrm{d} t} =\dfrac{-1.3}{6} rad/s[/tex]
So the angle's rate of change when the bottom of the ladder is 8ft from the wall is[tex]\dfrac{-1.3}{6} rad/s[/tex].
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Related Questions
can someone please tell me the answer to theses 2 questions
Answers
Sarah has read aloud in class 3 more times than Joel. Sarah has read 9 times. Which equation represents this situation?
A j + 3 = 9
B j − 9 = 3
C 3j = 9
D j − 3 = 9
Answers
9-3=6
j=6
6+3=9
A teacher made a pair of foam dice to use in math games. Each cube measured 10 in. On a side. How many square inches of fabric were needed to cover the two cubes?
Answers
Final answer:
A teacher will require 1200 square inches of fabric to cover the surface area of two foam cubes, with each side measuring 10 inches.
Explanation:
The question involves finding the total surface area of two foam cubes to determine how much fabric a teacher needs to cover both cubes. To solve this, we must calculate the surface area of one cube and then multiply by two because there are two identical cubes. Each face of a cube measures 10 inches on a side.
The area of one face is given by:
Area of one face = side × side = 10 in × 10 in = 100 square inches
Since a cube has 6 faces, the total surface area of one cube would be:
Total surface area of one cube = 6 × Area of one face = 6 × 100 square inches = 600 square inches
Therefore, for two cubes, we need:
Total surface area for two cubes = 2 × 600 square inches = 1200 square inches
The teacher will require 1200 square inches of fabric to cover the two cubes.
The area of a sector of a circle is given by the equation , 3.142S/360 where r is the radius of the circle and S is the angle measure of the sector. If Mia solved this equation for S, which of the following equations did she write?
Answers
A=3.142r² S/360
where:
r is the radius
S is the angle measure of the sector
If Mia solved the equation for S, the equations that she wrote would be found as follows;
From A=3.142r² S/360
making S the subject by first dividing both sides by 3.142r^2 we shall have:
A/3.142r²=S/360
next we multiply both sides by 360
360×A/3.142r²=S/360×360
this will give us
(360A)/(3.142r²)=S
therefore the equation that Mia will get is:
S=(360A)/(3.142r²)
Answer:
C.
Step-by-step explanation:
The diameter of small nerf balls manufactured at a factory in china is expected to be approximately normally distributed with a mean of 5.2 inches and a standard deviation of .08 inches. suppose a random sample of 20 balls is selected. find the interval that contains 95.44 percent of the sample means.
Answers
Answer:
The interval that contains 95.44 percent of the sample means is between 5.1642 inches and 5.2358 inches
Step-by-step explanation:
We need to understand the normal probability distribution and the central limit theorem to solve this question.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
[tex]\mu = 5.2, \sigma = 0.08, n = 20, s = \frac{0.08}{\sqrt{20}} = 0.0179[/tex]
Find the interval that contains 95.44 percent of the sample means.
0.5 - (0.9544/2) = 0.0228
Pvalue of 0.0228 when Z = -2.
0.5 + (0.9544/2) = 0.9772
Pvalue of 0.9772 when Z = 2.
So the interval is from X when Z = -2 to X when Z = 2
Z = 2
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]2 = \frac{X - 5.2}{0.0179}[/tex]
[tex]X - 5.2 = 2*0.0179[/tex]
[tex]X = 5.2358[/tex]
Z = -2
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]-2 = \frac{X - 5.2}{0.0179}[/tex]
[tex]X - 5.2 = -2*0.0179[/tex]
[tex]X = 5.1642[/tex]
The interval that contains 95.44 percent of the sample means is between 5.1642 inches and 5.2358 inches
What is 1/x divided by 1/y equal to ?
Answers
Final answer:
To calculate 1/x divided by 1/y, you multiply 1/x by the reciprocal of 1/y, which is y, resulting in y/x.
Explanation:
The question asks us to find the value of 1/x divided by 1/y. This is a division problem involving fractions and can be solved using the rules of division for fractions.
To divide one fraction by another, we multiply the first fraction by the reciprocal of the second. In this case, the reciprocal of 1/y is y/1 which simplifies to y. Therefore, the division 1/x ÷ 1/y becomes 1/x × y.
When we multiply these together, we get:
So, the answer is y/x.
Which conversion requires division? Use the metric table to help answer the question.
Answers
Like Centimeter to Meter
Like Meter to Kilometer
Like Feet to Mile
Like Yard to Mile
Like Feet to Yard
12000 feet divided by 3 is 4000, so 4000 yards, that kid of thing
Answer:
Decigrams to Decagrams/Dekagrams
Step-by-step explanation:
1 decigram is equal to 0.01 Deca/Dekagrams
Decigrams have less value than Decagrams.
Therefore,
You divide 1 decigram by 100 to get 0.01 decagrams.
The diagram below shows a process in earths crust which of the following statements best describes the process in the diagram
Answers
The layer with a plastic nature is the mantle (Layer C). The rocks in the mantle can deform and flow over long periods due to heat and pressure, giving them a plastic-like quality. This deformation is crucial for the movement of tectonic plates and various geological processes on Earth.
The Earth is composed of several layers, each with distinct properties and compositions. Starting from the center and moving outward, the major layers are:
Inner Core (Layer A):
The innermost layer is solid and composed mainly of iron and nickel. It is extremely hot, with temperatures reaching up to 9,000 degrees Fahrenheit (5,000 degrees Celsius). Despite the high temperature, the inner core remains solid due to intense pressure.
Outer Core (Layer B):
Surrounding the inner core is the outer core, which is also composed of molten iron and nickel. However, in the outer core, these materials are in a liquid state. The movement of the molten metal in this layer generates the Earth's magnetic field through the geodynamo process.
Mantle (Layer C):
Above the outer core is the mantle, a thick layer of solid rock that can flow slowly over geological timescales. The mantle is mostly composed of silicate minerals rich in iron and magnesium. The convection currents in the mantle are responsible for the movement of tectonic plates.
Crust (Layer D):
The outermost layer is the Earth's crust, which is relatively thin compared to the other layers. The crust is composed of solid rock and is divided into two types: the continental crust (found beneath continents) and the oceanic crust (found beneath oceans). The crust is where we find the solid ground we walk on.
The layer with a plastic nature is the mantle (Layer C). The rocks in the mantle can deform and flow over long periods due to heat and pressure, giving them a plastic-like quality. This deformation is crucial for the movement of tectonic plates and various geological processes on Earth.
The complete question is:
The diagram below shows four layers of the Earth.
Source:
Which of these layers is composed of materials with a plastic nature?
Layer A
Layer B
Layer C
Layer D
The layer with a plastic nature is the mantle (Layer C). The rocks in the mantle can deform and flow over long periods due to heat and pressure, giving them a plastic-like quality. This deformation is crucial for the movement of tectonic plates and various geological processes on Earth.
The Earth is composed of several layers, each with distinct properties and compositions. Starting from the center and moving outward, the major layers are:
Inner Core (Layer A):
The innermost layer is solid and composed mainly of iron and nickel. It is extremely hot, with temperatures reaching up to 9,000 degrees Fahrenheit (5,000 degrees Celsius). Despite the high temperature, the inner core remains solid due to intense pressure.
Outer Core (Layer B):
Surrounding the inner core is the outer core, which is also composed of molten iron and nickel. However, in the outer core, these materials are in a liquid state. The movement of the molten metal in this layer generates the Earth's magnetic field through the geodynamo process.
Mantle (Layer C):
Above the outer core is the mantle, a thick layer of solid rock that can flow slowly over geological timescales. The mantle is mostly composed of silicate minerals rich in iron and magnesium. The convection currents in the mantle are responsible for the movement of tectonic plates.
Crust (Layer D):
The outermost layer is the Earth's crust, which is relatively thin compared to the other layers. The crust is composed of solid rock and is divided into two types: the continental crust (found beneath continents) and the oceanic crust (found beneath oceans). The crust is where we find the solid ground we walk on.
The layer with a plastic nature is the mantle (Layer C). The rocks in the mantle can deform and flow over long periods due to heat and pressure, giving them a plastic-like quality. This deformation is crucial for the movement of tectonic plates and various geological processes on Earth.
The question probable maybe: The diagram below shows four layers of the Earth.
Which of these layers is composed of materials with a plastic nature?
Layer A
Layer B
Layer C
Layer D
If $7300 is invested in a savings account for which interest is compounded monthly, and if the $7300 turns into $8600 in 2 years, what is the interest rate of the savings account?
Answers
Answer:
8.22%
Step-by-step explanation:
The interest rate of the savings account is 8.22%
What is compound interest formula?[tex]A= P(1+\frac{r}{n} )^{nt}[/tex]
where, A : Amount
P = Principal amount
r = Annual interest rate as a decimal
R = Annual interest rate as a percent
r = R/100
n = number of compounding periods per unit of time
t = time in decimal years
What is the formula to find the interest rate?[tex]r=n[(\frac{A}{P} )^{\frac{1}{nt} }-1][/tex]
For given question,
the principal amount (P) = $7300
the amount (A) = $8600
period (t) = 2
n = 12 (since it's monthly compounding)
To find 'r'
Using the formula of rate of interest,
[tex]\Rightarrow r=n[(\frac{A}{P} )^{\frac{1}{nt} }-1]\\\\\Rightarrow r=12\times [(\frac{8600}{7300} )^{\frac{1}{12\times 2} }-1]\\\\\Rightarrow r=12\times [(1.18)^{\frac{1}{24} }-1]\\\\\Rightarrow \bold{r=0.0822}[/tex]
The rate of interest in percentage form,
[tex]\Rightarrow R=r\times 100\\\\\Rightarrow R=0.0822\times 100\\\\\Rightarrow \bold{R=8.22\%}[/tex]
Therefore, the interest rate of the savings account is 8.22%
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Ship A receives a distress signal from the east, and ship B receives a distress signal from the same vessel from the northwest. At what location is the vessel in distress located? Describe how you arrived at your conclusion using complete sentences. You must show all work in order to receive credit.
Answers
See the attached figure.
E direction is a line which is parallel to x-axis. (blue line).
NW direction lies on a line that is at an angle of 135 degrees to the positive direction of the x axis. (red lines).
Intersection between red line and blue line will give the coordinates of the vessel in distress location which is point C (3,2)
PLEASE HELP ME WITH THIS!!! Also make sure I can understand it it is also preferable to post a picture but either one will do thank you. (NEVER MIND I GOT IT IF YOU NEED IT JUST TELL ME)
Answers
In the figure attached, the first point is (2,5), you need to move 2 on the axis "speed (m/s)" and 5 on the axis "distance (m)". This means that the first point is the intersection between 2 m/s and 5 m. The second point is (3,8), then, you need to move 3 on the axis "speed (m/s)" an 8 on the axis "distance (m)" You must do this with the other values.
solve 5x<-10
help plz
Answers
To solve this equation, you divide both sides by 5.
This will give you x < -2.
I hope this helps!
Mr.EQ
15a^2 + 14a -8 use A.C method
Answers
A C method : you multiply the A and C -> find the factors of AC that can add up to B. (a² + bx + c)
15a² + 14a - 8
you multiply 15 and -8 = -120
Find the two factors of -120 that adds up to 14.
We could used 20 and -6.
So we can split the original equation to this:
15a² + 20a - 6a - 8
Now, we use grouping method.
Let's group 15a² + 20a and -6a - 8.
15a² + 20a can be factored like this :
5a(3a + 4)
-6a - 8 can be factored like this :
-2(3a + 4)
So it is 5a(3a + 4) × -2(3a + 4)
We can simplify this into this :
(5a - 2)(3a + 4)
So your final answer is (5a - 2)(3a + 4).
Hope this helped! :)
the length of rectangular garden is 5 feet longer than the width. If the are of the garden is 50 square feet, find the length and the width of the garden
Answers
length = x + 5
x(x + 5) = 50
x^2 + 5x = 50
x^2 + 5x - 50 = 0
(x + 10)(x - 5) = 0
x = 5 or - 10 (Rejected, length cannot be negative)
x = 5
x + 5 = 5 + 5 = 10
Therefore the width is 5 feet and the length is 10 feet
Final answer:
To solve for the length and width of the garden, we set up a quadratic equation based on the given area and the relation between length and width. Upon solving the quadratic equation, we find the width to be 5 feet and the length to be 10 feet.
Explanation:
The problem states that the length of a rectangular garden is 5 feet longer than its width and that the area of the garden is 50 square feet.
To find the length and width, let's define the width as w feet. Therefore, the length will be w + 5 feet. The area of a rectangle is found by multiplying the length and width, so we have the equation:
w × (w + 5) = 50
Expanding this we get:
w^2 + 5w - 50 = 0
Now we need to solve this quadratic equation for w. Factoring the quadratic, we find that (w + 10)(w - 5) = 0, which gives us two possible solutions for w: -10 and 5.
Since a width can't be negative, we discard -10 and take w = 5 feet. So, the width of the garden is 5 feet and the length is 5 + 5 = 10 feet.
Which is the correct input-output table for the function ?f(x)=3/x+4 HELP what are the table answers
Answers
Answer:
The correct tabe is here,
let take f(x)=y
then if
x=1,f(x)=7
x=2,f(x)=11/2
x=3,f(x)=5
x=4,f(x)=19/4
and so on so so far
Please help!!!!:
Mrs. Greenhaus has a six-sided number cube and a spinner. The number cube has the numbers 1 through 6 on its faces, and the spinner has four equal regions with blue, red, yellow, and green each in a different region. This tree diagram represents the outcomes of rolling the number cube and then spinning the spinner.
What is the probability of rolling an even number and landing on blue?
A.1/2
B.7/8
C.1/4
D.1/8
Answers
What is the equation of the line?
Answers
y = 0 . . . . is the equation of the line.
_____
The line shows y to have the same value for every value of x. Thus the value of y is a constant, and there is no dependence on x at all. This is true for any horizontal line. Here, the constant value is zero, so the equation is ...
... y = 0
Last year, a person wrote 134134 checks. let the random variable x represent the number of checks he wrote in one day, and assume that it has a poisson distribution. what is the mean number of checks written per day? what is the standard deviation? what is the variance?
Answers
(a) mean = 367.24
(b) standard deviation = 19.17
(c) variance = 367.24
Explanations:
(a) Let [tex]\lambda[/tex] = rate parameter of parameter, which is the rate at which the number of checks are made in one day.
Since the person wrote 134,134 checks in last year, we can compute the rate parameter by dividing 134,134 by 365.251 because there are 365.251 days in a year. So,
[tex]\lambda = \frac{134,134}{365.251} \approx 367.237872[/tex]
Since the number of checks in one day follows a Poisson distribution, the mean number of checks in one day is the rate parameter [tex]\lambda[/tex]. Hence, the mean number of checks in one day is approximately 367.24.
(b) Since the number of checks in one day follows a Poisson distribution, the variance is equal to the rate parameter. Since the standard deviation is the square root of variance, the standard deviation is the square root of the rate parameter. In terms of equations,
[tex]\text{standard deviation} \\ = \sqrt{\lambda} \\ = \sqrt{\frac{134,134}{365.251}} \\ \boxed{\text{standard deviation} \approx 19.17}[/tex]
(c) As mentioned in (b), because the number of checks written per day follows poisson distribution, the variance is equal to the rate parameter. Since the rate parameter is approximately 367.24, therefore the variance is approximately 367.24.
What would happen to a monthly payment if the interest rate increased
Answers
What is the value of x?
#2
a. 71
b. 142
c. 152
d. 76
#3
a. 68
b. 62
c. 112
d. 124
Answers
wendy weighs 8lb less than jean if jeans weight is p lb then what is an appropriate expression fir wendys weight
Answers
Since Wendy is 8 pounds less than Jean, we take Jean's weight and subtract 8. If Jean's weight is p, then Wendy's would be p-8.
Answer:
p-8
Step-by-step explanation:
Round 9.098 to the nearest tenth.
Answers
Final answer:
To round 9.098 to the nearest tenth, we round up because the hundredths digit is 9, resulting in 9.1.
Explanation:
Rounding 9.098 to the nearest tenth:
Identify the tenths place in 9.098, which is 0.1.The digit immediately after the tenths place is 9, which is 5 or higher, so you need to round up.Therefore, when rounding 9.098 to the nearest tenth, the result is 9.1.What’s 3 1/3 times 4 3/4
Answers
The answer is "95/6".
Reason:
First write the answers into improper fraction:
3 1/3=10/3
4 3/4=19/4
10*19=190
3*4=12
190/2=95
12/2=6
=95/6
If you need anymore help feel free to ask me!
Hope this helps!
~Nonportrit
I hope You are having a good day!
I would be happy to help!
Your question states: What’s 3 1/3 times 4 3/4
This can also be written as: 3 1/3 * 4 3/4 or, to make it easier, you can write it as 10/3 * 19/4
For this, I will write it as 10/3 * 19/4
The first step to solving this is find a common denominator.
The common denominator will be 12.
to make 10/3 have a denominator of 12, we must multiply the top and the bottom (the numerator and denominator) by 4
4 * 10 = 40
4 * 3 = 12
It is now 10/12
To make 19/4 have a denominator of 12, we must multiply the top and the bottom by 3.
3 * 19 = 247
3 * 4 = 12
Now we have 247/12
Now we will multiply
[tex] \frac{10 * 247}{12 * 12} [/tex]
2470/144
we can still simplify this
1235/72
Hope I helped!
Let me know if you need anything else!
~ Zoe
The formula for finding the present value of an item that depreciates yearly is v = c - crt. In this formula, v is the present value, c is the original cost, r is the rate of depreciation per year, and t is the number of years that have passed. After 7 years, what is the value of a car originally costing $27,000 that depreciated at a rate of 0.1 per year
Answers
Using the formula and substituting our known information, we have:
v = 27000 - 27000(0.1)(7)
v = 27000 - 18900
v = 8100
Final answer:
To calculate the present value of a car after 7 years, using the formula v = c - crt with the given values, the present value is $8,100.
Explanation:
The question involves finding the present value of a depreciating asset, in this case, a car. Using the provided formula v = c - crt where v is the present value, c is the original cost, r is the rate of depreciation per year, and t is the number of years, we can calculate the value of a car after a certain time period.
Given that the original cost c is $27,000, the rate of depreciation r is 0.1 (or 10%), and the time t is 7 years, we can substitute these values into the formula to find the present value of the car after 7 years. The calculation is as follows: v = $27,000 - ($27,000 × 0.1 × 7). Simplifying, we get: v = $27,000 - ($27,000 × 0.7) = $27,000 - $18,900 = $8,100.
Factor
25m^100 − 121n^16
1. (5m^50 - 11n^8)^2
2. (5m^10 - 11n^4) (5m^10 + 11n^4)
3. (5m^50 - 11n^8) (5m^50 + 11n^8)
4. (5m^10 - 11n^4)^2
Answers
The correct answer is C) (5m^50 - 11n^8) (5m^50 + 11n^8)
We can tell this because of the rule regarding factoring the difference of two perfect squares. When we have two squares being multiplied, we can use the following rule.
a^2 - b^2 = (a - b)(a + b)
In this case, or first term is 25m^100. So we can solve that by setting it equal to a^2.
a^2 = 25m^100 -----> take the square root of both sides
a = 5m^50
Then we can do the same for the b term.
b^2 = 121n^16 ----->take the square root of both sides
b = 11n^8
Now we can use both in the equation already given
(a - b)(a + b)
(5m^50 - 11n^16)(5m^50 + 11n^16)
Without doing any computation, decide which has a higher probability, assuming each sample is from a population that is normally distributed with a mean equal to 100 and a standard deviation equal to 15. explain your reasoning. (a) p(90≤sample mean≤110) for a random sample of size n = 10 (b) p(90≤sample mean≤110) for a random sample of size n = 20
Answers
Explanation:
Since we're computing for the probability of the sample mean, we consider the z-score and the standard deviation of the sampling distribution. Recall that the standard deviation of the sampling distribution approximately the quotient of the population standard deviation and the square root of the sample size.
So, if the sample size higher, the standard deviation of the sampling distribution is lower. Since the sample size in (b) is higher, the standard deviation of the sampling distribution in (b) is lower.
Moreover, since the mean of the sampling distribution is the same as the population mean, the lower the standard deviation, the wider the range of z-scores. Because the standard deviation in (b) is lower, it has a wider range of z-scores.
Note that in a normal distribution, if the probability has wider range of z-scores, it has a higher probability. Therefore, the probability in (b) has higher probability than the probability in (a) because it has wider range of z-scores than the probability in (a).
To decide which random sample has a higher probability, we need to consider the sample size. As the sample size increases, the sample mean tends to approach the population mean more closely. We can use the properties of the normal distribution and the formula for the standard error of the mean to calculate the probabilities for each random sample size.
Explanation:To decide which has a higher probability, we need to consider the sample size of each random sample. In general, as the sample size increases, the sample mean tends to approach the population mean more closely.
Given that each sample is from a population that is normally distributed with a mean of 100 and a standard deviation of 15, we can use the properties of the normal distribution to calculate the probabilities.
(a) For a random sample of size n = 10, the probability of the sample mean being between 90 and 110 can be calculated using the formula for the standard error of the mean:
Standard error of the mean = standard deviation / sqrt(sample size)
Using this formula, the standard error of the mean for n = 10 is 15 / sqrt(10) = 4.74.
Since the sample mean follows a normal distribution, we can use the properties of the normal distribution to find the probability:
p(90 ≤ sample mean ≤ 110) = p((90 - 100) / 4.74 ≤ (sample mean - 100) / 4.74 ≤ (110 - 100) / 4.74)
We can look up the probabilities for the z-scores using a standard normal distribution table or a calculator to find the probability for the given range.
(b) For a random sample of size n = 20, we can follow the same steps as above to calculate the probability of the sample mean being between 90 and 110.
The standard error of the mean for n = 20 is 15 / sqrt(20) = 3.35.
p(90 ≤ sample mean ≤ 110) = p((90 - 100) / 3.35 ≤ (sample mean - 100) / 3.35 ≤ (110 - 100) / 3.35)
We can again use a standard normal distribution table or a calculator to find the probability for the given range.
Lucy has been given a list of 5
bands and asked to place a vote. Her vote must have the names of her favorite and second favorite bands from the list. How many different votes are possible?
Answers
Evaluate the expression 42+6⋅52−33÷32.
Answers
= 42+6⋅52−33÷32
multiply first
= 42 + (6*52) - 33 ÷ 32
= 42 + 312 - 33 ÷ 32
division second
= 42 + 312 - 1.03125
addition next
= 354 - 1.03125
subtraction last
= 352.96875
ANSWER: 352.96875 (*if rounding, answer is 353)
Hope this helps! :)
What percent decrease occurs when a stock goes from 6.50 per share to 4.25 per share?
Answers
To find the percent decrease, find the amount of decrease: 6.50 - 4.25 = 2.25. Now divide this by the original price of the stock:
2.25/6.50 = 0.3462 ≈ 0.35
Suppose heights of 20-year-old men are approximately normally distributed with a mean of 71 in. and a population standard deviation of 5 in. a random sample of twenty 20-year-old men is selected and measured. find the probability that the sample mean x lies between 68 and 74 inches.