To minimize the cost of a rectangular milk carton that holds 128 cubic cm, we need to calculate the dimensions that minimize the surface area cost. By setting up an optimization problem and using calculus, we can find the values of length, width, and height that satisfy the volume constraint and result in the lowest cost.
Minimizing Cost for a Rectangular Milk Carton
To find the dimensions of a rectangular milk carton with a given volume that minimize the cost of materials used, we can set up an optimization problem using calculus. First, we know the volume of the milk carton must be $$128 cm^3$, which gives us the constraint:
V = lwh = 128
Next, we need to express the cost function in terms of the dimensions of the box. The sides of the box cost $$1 cent per square cm, while the top and bottom cost $2 cents per square cm. Thus, the total cost, C, in cents, is:
C = 2lw + 2wh + 2lh + (4 * l * w)
To minimize the cost, we would take the partial derivatives with respect to l, w, and h, set them equal to zero, and solve the system of equations while taking into account the volume constraint. This involves the method of Lagrange multipliers or directly substituting the volume constraint into the cost function to eliminate one variable and then taking the derivative with respect to the other variables.
By finding the derivative of the cost function and setting it to zero, you can determine the values of l, w, and h that will result in the minimum cost while respecting the volume constraint. Since this is an applied problem, it is important to check that the resulting values are practical, meaning they should be positive and make sense for a milk carton.
Angelica has been depositing $280 each month into a savings account with an APR of 2.76% for the last 3 years. If she continues depositing this amount for an additional 12 years, what will the balance in her savings account be?
Answer:
$62,490.65
Step-by-step explanation:
If we assume her deposits are at the beginning of the month, and that the interest is compounded monthly, the future value is that of an "annuity due." The formula is ...
FV = P(1+r/n)((1+r/n)^(nt)-1)/(r/n)
where r is the APR (.0276), n is the number of yearly compoundings (12), P is the monthly payment ($280), and t is the number of years (15). Putting the numbers into the formula and doing the arithmetic, we get ...
FV = $280(1.0023)(1.0023^180 -1)/(.0023) ≈ $62,490.65
Angelica's account balance after 15 years will be $62,490.65.
_____
If her deposits are at the end of the month, the balance will be $62,347.25.
You select a family with three children. If M represents a male child, and F represents a female child, the set of equally likely outcomes for the children's genders is { MMM, MMF, FFM, MFF, FFM, FMF, FFM, FFF }. Find the probability of selecting a family with nothing 6 male children.
Answer:
1/8
Step-by-step explanation:
This is actually pretty easy.
First, you have the sample of the children.
You know that the family with 3 children, can only have the following children:
MMM: all males
MMF: 2 males, 1 female
MFF: 1 male, 2 females
MFM: 2 males, 1 female (different order)
FMM: 1 female, 2 males
FFM: 2 females, 1 male
FMF: 2 females, 1 male
FFF: 3 females
If you count all of these possibilities, we have 8 possible cases of family with the childrens.
In only one of them, we have only females and no males, which is the last one, all 3 females.
Therefore, the probability to select a family with no male, is only 1/8.
In the set of all possible sets of genders for three children, which is {MMM, MMF, MFF, FMM, FFM, FMF, MFM, FFF}, the event of having no male children is 'FFF'. As there are 8 possible outcomes in total, the probability of selecting a family with no male children is 1/8, or 0.125.
Explanation:It seems like there is a typo in your question. It asks about the probability of selecting a family with 'nothing 6 male children', which doesn't quite make sense. However, if the question is to find the probability of selecting a family with no male children, we can certainly answer that. In your given set of outcomes, { MMM, MMF, FFM, MFF, FFM, FMF, FFM, FFF }, there seems to be a mistake as FFM is repeated three times. The correct set of equally likely outcomes for a family with three children should be { MMM, MMF, MFF, FMM, FFM, FMF, MFM, FFF }.
Each outcome is equally likely, hence, the probability of any particular outcome is 1 divided by the total number of outcomes. The total unique outcomes for the genders of three children is 2^3, or 8. The event of having no male child corresponds to the outcome 'FFF'. Since there is only one such outcome, the probability of a family with no male children is 1/8 or 0.125.
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Supppose A is an invertible n\times n matrix and v is an eigenvector of A with associated eigenvalue -3. Convince yourself that v is an eigenvalue of the following matrices, and find the associated eigenvalues:
1. \ A^6, \ eigenvalue = ,
2. \ A^{-1}, \ eigenvalue = ,
3. \ A + 5 I_n, \ eigenvalue = ,
4. \ 6 A, \ eigenvalue = .
Answer:you will an app go to =567
Is 27/50 closer to 1/2, 1 or 0
Answer:
I think it's closer to [tex]\frac{1}{2}[/tex]
Step-by-step explanation:
[tex]\frac{27}{50} = 0.54\\[/tex]\\
i do know that 1/2 is < 27/50
The volume of a sphere is increasing at a constant rate of 141 cubic feet per minute. At the instant when the radius of the sphere is 11 feet, what is the rate of change of the radius? The volume of a sphere can be found with the equation V=4/3pi r^3 . Round your answer to three decimal places.
Answer:
0.093 ft/min
Step-by-step explanation:
V = 4/3 π r³
Take derivative with respect to time:
dV/dt = 4π r² dr/dt
Plug in values:
141 = 4π (11)² dr/dt
dr/dt = 141 / (484π)
dr/dt ≈ 0.093
The radius is increasing at 0.093 ft/min.
The increasing rate of the radius is equal to 0.093 ft/min.
What is the volume?Volume is defined as the space occupied by the object in a three-dimensional space. The sphere is the shape of a circular ball.
The volume is calculated by the formula below,
V = 4/3 π r³
Take the derivative with respect to time:
dV/dt = 4π r² dr/dt
Solve the equation for the rate of change of radius of the sphere,
141 = 4π (11)² dr/dt
dr/dt = 141 / (484π)
dr/dt ≈ 0.093
Therefore, the increasing rate of the radius is equal to 0.093 ft/min.
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help me QUICK, plz! 15 points!
Answer:
t=8
Step-by-step explanation:
Subtract 12t from both sides
(12t-12t) + 16 = (13t-12t) + 24
16 = 1t (or just t) + 24
Then subtract 16 from both sides, and bring "t" over the equals sign making it a negative integer.
-t = 8
t = 8
Brandon is an amateur marksman. When he takes aim at a particular target on the shooting range, there is a 0.1, point, probability that he will hit it. One day, Brandon decides to attempt to hit 10 such targets in a row.
Assuming that Brandon is equally likely to hit each of the 10 targets, what is the probability that he will hit at least one of them?
Answer:
.65
Step-by-step explanation:
Strategy:
In this situation it is much easier to calculate the probability of the event we are looking for (he hits at least one target) by calculating the probability of its complement (he misses every target), and subtracting from 1.
In other words, we can use this strategy:
P(at least one hit)=1-P(miss all 10)
Calculations:
P(at least one hit)
=1-P(miss all 10)
=1-(0.9)^10
≈1-0.349
≈0.65
Answer:
P(at least one hit)≈0.65
I hope this helps!!!
The probability that Brandon will hit at least one of them is 0.65 approx.
How to find that a given condition can be modeled by binomial distribution?Binomial distributions consists of n independent Bernoulli trials.
Bernoulli trials are those trials which end up randomly either on success (with probability p) or on failures( with probability 1- p = q (say))
Suppose we have random variable X pertaining binomial distribution with parameters n and p, then it is written as
[tex]X \sim B(n,p)[/tex]
The probability that out of n trials, there'd be x successes is given by
[tex]P(X =x) = \: ^nC_xp^x(1-p)^{n-x}[/tex]
For the considered case, as each hit is independent of each other, and there is either hit or not hit, so this situation can be modeled by binomial distribution.
For this case, we get:
n = 10p = probability of success(we call it success if hit occurs) = 0.1X = count of successes in those 10 attemptsThen, we get: [tex]X \sim B(n=10,p=0.1)[/tex]
The probability that Brandon will hit at least one of them is written symbolically as:
[tex]P(X \geq 1)[/tex]
We can rewrite it as:
[tex]P(X \geq 1) = 1 - P(X < 1) = 1 - P(X = 0)[/tex]
Using the probability function of binomial distribution, we get:
[tex]P(X \geq 1) = 1 - \: ^{10}C_0(0.1)^0(1-0.1)^{10} = 1 -0.9^{10} \approx 0.65[/tex]
Thus, the probability that Brandon will hit at least one of them is 0.65 approx.
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Given the functionf ( x ) = x^2 + 7 x + 10/ x^2 + 9 x + 20
Describe where the function has a vertical asymptote and how you found your answer. Remember that an asymptote is represented by an equation of a line and not just a single value.
x = -4 is a vertical asymptote for the function.
Explanation:The graph of [tex]y=f(x)[/tex] is a vertical has an asymptote at [tex]x=a[/tex] if at least one of the following statements is true:
[tex]1) \ \underset{x\rightarrow a^{-}}{lim}f(x)=\infty\\ \\ 2) \ \underset{x\rightarrow a^{-}}{lim}f(x)=-\infty \\ \\ 3) \ \underset{x\rightarrow a^{+}}{lim}f(x)=\infty \\ \\ 4) \ \underset{x\rightarrow a^{+}}{lim}f(x)=\infty[/tex]
The function is:
[tex]f(x)=\frac{x^2+7x+10}{x^2+9x+20}[/tex]
First of all, let't factor out:
[tex]f(x)=\frac{x^2+5x+2x+10}{x^2+5x+4x+20} \\ \\ f(x)=\frac{x(x+5)+2(x+5)}{x(x+5)+4(x+5)} \\ \\ f(x)=\frac{(x+5)(x+2)}{(x+5)(x+4)} \\ \\ f(x)=\frac{(x+2)}{(x+4)}, \ x\neq 5[/tex]
From here:
[tex]\bullet \ When \ x \ approaches \ -4 \ on \ the \ right: \\ \\ \underset{x\rightarrow -4^{+}}{lim}\frac{(x+2)}{(x+4)}=? \\ \\ \underset{x\rightarrow -4^{+}}{lim}\frac{(-4^{+}+2)}{(-4^{+}+4)} \\ \\ \\ The \ numerator \ is \ negative \ and \ the \ denominator \\ is \ a \ small \ positive \ number. \ So: \\ \\ \underset{x\rightarrow -4^{+}}{lim}\frac{(x+2)}{(x+4)}=-\infty[/tex]
[tex]\bullet \ When \ x \ approaches \ -4 \ on \ the \ left: \\ \\ \underset{x\rightarrow -4^{-}}{lim}\frac{(x+2)}{(x+4)}=? \\ \\ \underset{x\rightarrow -4^{-}}{lim}\frac{(-4^{-}+2)}{(-4^{-}+4)} \\ \\ \\ The \ numerator \ is \ a \ negative \ and \ the \ denominator \\ is \ a \ small \ negative \ number \ too. \ So: \\ \\ \underset{x\rightarrow -4^{-}}{lim}\frac{(x+2)}{(x+4)}=+\infty[/tex]
Accordingly:
[tex]x=-4 \ is \ a \ vertical \ asymptote \ for \\ \\ f(x)=\frac{x^2+5x+2x+10}{x^2+5x+4x+20}[/tex]
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Please help me! Picture below
Answer:
A i think
Step-by-step explanation:
5:3 =n:7 Solve for n.
(there fractions but i have to write them with : because i cant write fractions)
[tex] \frac{5}{3} = \frac{x}{7} [/tex]
you them cross multiply
5×7=35
3×X= 3x
35=3x
x(n)= 11.6
Answer:
n = 35/3 or 11 2/3.
Step-by-step explanation:
5/3 = n/7
Cross multiply:
3n = 5*7
3n = 35
n = 35/3
Assume a circle of radius r has the same area as a square with side length s. Express the radius of the circle in terms of the length of a side of the square (i.e. write r as a function of s)
Final answer:
To express the radius of a circle in terms of the length of a side of a square, use the formula: r = s/√π.
Explanation:
A circle of radius r has the same area as a square with side length s. We can express the radius of the circle in terms of the length of a side of the square using the formula for the area of a circle: A = πr², where π is approximately 3.14159. Similarly, for the square, the area is given by A = s². Equating these two areas gives us the equation:
s² = πr²
Now, we can solve for r by rearranging the equation:
r = s/√π
Final answer:
To express the radius of a circle in terms of the side length of an equal-area square, use the formula r = s / √π.
Explanation:
To express the radius r of a circle in terms of the length of a side of the square s, where the circle and square have the same area, we start by setting the area formulas for both shapes equal to each other. The area of the circle is πr^2 and the area of the square is s^2. By equating the two areas, we have πr^2 = s^2. To solve for r, we need to take the square root of both sides of the equation divided by π, which gives us r = s / √π.
Katie keeps a record of her transactions in a notebook. Looking over her records for the past two weeks, Katie finds that her bank statement does not match her notebook, and she actually has more money than she thought she did. These are the relevant entries in Katie’s notebook:
Transaction
Debit ($)
Credit ($)
Beginning Balance
889.23
Fence repair
75.86
---
Dry cleaning
32.38
---
Paycheck
---
140.91
Gift from cousin
---
35.00
Car payment
178.66
---
Tax refund
---
66.18
New chair
161.55
---
Ending Balance
682.87
However, Katie’s bank statement says that her ending balance is $707.10. Which of the following is a possible explanation for the discrepancy?
a.
Katie’s neighbor paid for half of the fence repair, which Katie forgot to record.
b.
Katie transposed the digits in the cost of the dry cleaning, which actually cost $23.38.
c.
Katie’s new chair was on sale for 15% off, making it cost $137.32, but Katie wrote down the before-sale price.
d.
Katie’s tax refund came in two equal-sized checks, but Katie only recorded one of them.
Answer: C
Step-by-step explanation:707.10-682.87= 24.23. Similar difference can also be found in only chair which is. 161.55-137.32=24.23
Answer:
The correct answer is C on edge :) i just did the test
Step-by-step explanation:
A model is made of a car. The car is 3 meters long and the model is 3 centimeters long. What is the ratio of the length of the car to the length of the model? A. 3 : 3 B. 1 : 100 C. 1 : 3 D. 100 : 1
Answer:
D
Step-by-step explanation:
a meter is 100 centimeters so the ratio of the real car to the model is 300 centimeters to 3 centimeters, or 100:1 so D
.---------. _
'-O------O--'
11. Solve the problem. A company manufactures televisions in batches of 25 and there is a 1% rate of defects. Find the standard deviation for the number of defects per batch.
0.7
0.9
0.5
72.8
Answer: 0.5
Step-by-step explanation:
For binary distribution with parameters p (probability of getting success in each trial) and n (Total trials) , we have
[tex]\sigma=\sqrt{np(1-p)}[/tex]
We are given that ,
Total batches of televisions : n=25
The probability of defects : p= 0.01
Here success is getting defective batch .
Then, the standard deviation for the number of defects per batch will be :-
[tex]\sigma=\sqrt{(25)(0.01)(1-0.01)}\\\\=\sqrt{(25)(0.01)(0.99)}\\\\=\sqrt{0.2475}\\\\=0.497493718553\approx0.5[/tex] [Rounedde to the nearest tenth.]
Therefore, the standard deviation for the number of defects per batch =0.5
Daniela invested a total of $50,000, some in a certificate of deposit (CD) and the remainder in bonds. The amount invested in bonds was $5,000 more than twice the amount she put into the CD. How much did she invest in each account? Call the amount that Daniela invested in the CD d and the amount she invested in bonds b.
Answer:
The amount invested in bonds = 35,000
The amount invested in CD = 15,000.
Step-by-step explanation:
The total amount that Daniela invested is $50,000, this means if we call the amount invested in bonds [tex]b[/tex], and the amount invested in CD [tex]d[/tex], then we have:
[tex]b+d=50,000[/tex] this says the total amount Daniela invested is $50,000.
And since the amount invested in bonds [tex]b[/tex] is $5,000 more than twice the amount Daniela put into the CD, we have:
[tex]b=5,000+2d[/tex].
Thus, we have two equations and two unknowns [tex]b[/tex] and [tex]d[/tex]:
(1). [tex]b+d=50,000[/tex]
(2). [tex]b=5,000+2d[/tex],
and we solve this system by substituting [tex]b[/tex] from the second equation into the first:
[tex]b+d=50,000\\5,000+2d+d=50,000\\3d=45,000\\\\\boxed{d=15,000}[/tex]
or, the amount invested in CD is $15,000.
With the value of [tex]d[/tex] in hand, we now solve for [tex]b[/tex] from equation(2):
[tex]b=5,000+2d\\b=5000+2(15,000)\\\boxed{b=35,000}[/tex]
or, the amount invested in bonds is $35,000.
Late shows Some TV shows begin after their scheduled times when earlier programs run late. According to a network’s records, about 3% of its shows start late. To find the probability that three consecutive shows on this network start on time, can we multiply (0.97)(0.97)(0.97)? Why or why not?
Answer:
No because the probability of consecutive shows starting late are not independent events
Step-by-step explanation:
Is good begin with the definition of independent events
When we say Independent Events we are refering to events which occur with no dependency of other evnts. Basically when the occurrence of one event is not affected by another one.
When two events are independent P(A and B) = P(A)xP(B)
But for this case we can't multiply 0.97x0.97x0.97 in order to find the probability that 3 consecutive shows start on time, because the probability for shows starting late are not independent events, because if the second show is late, the probability that the next show would be late is higher. And for this reason we can use the independency concept here and the multiplication of probabilities in order to find the probability required.
Yes, you can multiply (0.97)(0.97)(0.97) because the events are independent.
Yes, you can multiply (0.97)(0.97)(0.97) to find the probability that three consecutive shows on this network start on time. This is because the events are independent; the outcome of one show starting on time does not affect the outcome of the others.
Therefore, the probability that three consecutive shows start on time is the product of the probabilities of each show starting on time: P(all three shows start on time) = 0.97 * 0.97 * 0.97 = 0.912673.
Fresh pond has a population of 854 and is increasing by 3 people per year. Strawberry has a population of 427 and is increasing by 10% per year. Write an equation that models the growth for each town.
Answer:
Fresh Pond: p(t) = 854 +3tStrawberry: p(t) = 427·1.10^tStep-by-step explanation:
(a) The general term of an arithmetic sequence is ...
an = a1 + d(n -1)
If we let the sequence of population numbers be modeled by this, and we use t for the number of years, we want n=1 for t=0, so n = t+1 and we have ...
p(t) = 854 +3(t+1-1)
p(t) = 854 +3t
__
(b) The general term of a geometric sequence is ...
an = a1·r^(n-1)
were r is the common ratio. Here, the multiplier from one year to the next is 1+10% = 1.10. Again, n=t+1, so the population equation is ...
p(t) = 427·1.10^(t+1-1)
p(t) = 427·1.10^t
Scenario: The market for used cell phones is very popular in Barylia. However, several phones available in this market are of inferior quality and it is often impossible to differentiate between a good-quality phone and a poor-quality phone. Refer to the scenario above. Based on the given information, we can conclude that the market for used cell phones in Barylia: is perfectly competitive. has only one seller. is monopolistically competitive. has asymmetric information.
Answer:
D: has asymmetric information.
Step-by-step explanation:
Several phones available in Barylia market are of inferior quality and it is often impossible to differentiate between a good-quality phone and a poor-quality phone.
So, we can conclude that the market for used cell phones in Barylia: has asymmetric information.
Asymmetric information or often called information failure shows the decision in transactions where one market or party has got better information than the others.
Rover eats 3/4 of a can of cat food each day and Bono eats 1/2 of a can food each day.Cat food costs $5.00 for three cans.It is only sold in 3 can packs.How much does it cost for a 60 day supply of cat food?
Answer:
It would Cost $125 for a 60 day supply of cat food.
Step-by-step explanation:
Given:
Rover eats cat food each day = [tex]\frac{3}{4}[/tex] = 0.75 can
Bono eats cat food each day = [tex]\frac{1}{2}[/tex] = 0.5 can
Each day consumption for both cats = 0.75+0.5 = 1.25 can
Each day both cats consume = 1.25 cans
For 60 days both cats consume = Number of cans in 60 days.
By Using Unitary method we get;
Number of cans in 60 days = [tex]1.25\times60=75 \ cans[/tex]
Now Cans are sold in a pack of 3.
Hence we will divide number of cans in 60 days with 3 we get;
Number of can packs required for 60 days = [tex]\frac{75}{3}=25 \ can \ packs[/tex]
Now Cost for each Can packs(3 can) = $5.00
Hence Cost for 25 Can packs (75 cans) = Price for 25 can packs(75 can)
By Using Unitary method we get;
Price for 25 can packs (75 cans) = [tex]5\times 25 = \$125[/tex]
Hence Price for 25 can packs (75 cans) which are used for 60 supply of cat food is $125.
Recall the equation for a circle with center ( h , k ) and radius r . At what point in the first quadrant does the line with equation y = x + 1 intersect the circle with radius 6 and center (0, 1)?
Answer:
(3√2, 1+3√2) ≈ (4.243, 5.243)
Step-by-step explanation:
The line has a slope of 1 and goes through the center of the circle, so will intersect the circle at the point ...
(6, 6)/√2 +(0, 1) = (3√2, 1+3√2) ≈ (4.243, 5.243)
Jack went out for dinner at Red Lobster, his meal was $45, jack wants to leave an 18% tip, list three different calculations jack can use to determine his bill including the tip
Answer:
Step-by-step explanation:
Jack went out for dinner at Red Lobster and his meal was $45, This is the amount that he needs to pay if there was no tax or tip.
jack wants to leave an 18% tip, The amount of tip that jack wants to leave is 18/100 × 45 = 0.18×45 = $8.1
To determine how much jack needs to pay including the tip,
Amount that jack would pay is bill + tip = 45 + 8.1 = $53.1
Another way of calculating it is,
Since the tip is 18% and his bill was 100%, we will add the percentages and multiply by the bill. It becomes
118/100 × 45 = $53.1
(x^2–x–4)multiplied by(x–5)
Answer:
x³ -6x² +x +20
Step-by-step explanation:
The distributive property is useful for making sure you have all of the partial products.
(x^2–x–4)(x–5) = x(x^2–x–4) -5(x^2–x–4)
= (x^3 -x^2 -4x) +(-5x^2 +5x +20)
= x^3 +(-1-5)x^2 +(-4+5)x +20
= x^3 -6x^2 +x +20
Consider the sequence:
5, 7, 11, 19, 35,....
Write an explicit definition that defines the sequence:
Group of answer choices
a_n=2n+3 a n = 2 n + 3
a_n=3n+2 a n = 3 n + 2
a_n=3n^2 a n = 3 n 2
a_n=2^n+3
Answer:
[tex]a_{n}=2^{n}+3[/tex]
Step-by-step explanation:
The given sequence is not arithmetic, it's a geometric sequence, that means the sequence is obtain using powers. The faster way to find the answer is to try options that fist with a geometric sequence. If we try the last one, we'll find that's the answer.
We need to try for n=1, n=2, n=3, n=4 and n=5:
a_{1}=2^{1}+3=5
a_{2}=2^{2}+3=4+3=7
a_{3}=2^{3}+3=8+3=11
a_{4}=2^{4}+3=16+3=19
a_{5}=2^{5}+3=32+3=35
Therefore, the right answer is the last choice, because as you can observe, it fits perfectly with the given sequence.
Which ordered pairs are on the line with equation 3x-y=2.
a) (0, -2) b) (-3, 4) c) (1, -5)
The ordered pair (0 , -2) lies on the line with equation 3 x - y = 2
Step-by-step explanation:
To prove that a point lies on a line
Substitute x and y in the equation by the coordinates of the pointIf the two sides of the equation equal each other, then the point lies on the lineIf the two sides of the equation not equal each other then the point does't lie on the lineThe equation of the line is 3 x - y = 2
a) Point (0 , -2)
∵ x = 0 and y = -2
- Substitute the values of x and y in the left hand side
∵ The left hand side is 3 x - y
∵ 3(0) - (-2) = 0 + 2 = 2
∴ The left hand side = 2
∵ The right hand side = 2
∴ The two sides of the equation are equal
∴ The ordered pair (0 , -2) lies on the line
b) Point (-3 , 4)
∵ x = -3 and y = 4
- Substitute the values of x and y in the left hand side
∵ The left hand side is 3 x - y
∵ 3(-3) - (4) = -9 - 4 = -13
∴ The left hand side = -13
∵ The right hand side = 2
∴ The two sides of the equation are not equal
∴ The ordered pair (-3 , 4) doesn't lie on the line
c) Point (1 , -5)
∵ x = 1 and y = -5
- Substitute the values of x and y in the left hand side
∵ The left hand side is 3 x - y
∵ 3(1) - (-5) = 3 + 5 = 8
∴ The left hand side = 8
∵ The right hand side = 2
∴ The two sides of the equation are not equal
∴ The ordered pair (1 , -5) doesn't lie on the line
The ordered pair (0 , -2) lies on the line with equation 3 x - y = 2
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If a weight hanging on a string of length 5 feet swings through 6° on either side of the vertical, how long is the arc through which the weight moves from one high point to the next high point?\
Answer:
1.047 ft
Step-by-step explanation:
The length of an arc is given by ...
s = rθ
where s is the arc length, r is the radius, and θ is the central angle in radians. Your arc subtends an angle of 12° = (12·π/180) = π/15 radians. The length of the arc is then ...
s = (5 ft)(π/15) = π/3 ft ≈ 1.047 ft
The weight swings through a total angle of 12°, corresponding to an arc length of approximately 1.047 feet on the circumference of the circle with radius 5 feet.
Explanation:To answer this question, we need to understand that the weight swings through an arc, and this arc is a part of the circumference of a circle. Given the length of string (5 feet) is the radius, and the weight swings through 6° on either side of the vertical, we can calculate the total arc length.
Firstly, you should know that the total angle a circle encompasses is 360°. So, the weight swings through a total angle of 6° x 2 = 12°.
Secondly, recall the formula for the circumference of a circle is 2πr, or in our case 2π x 5 feet. Now, to find the length of the arc corresponding to 12°, we will use the proportion of the swing angle to the total angle, i.e., (12/360) x (2π x 5 feet).
In this way, the length of the arc travelled is almost 1.047 feet.
Learn more about Arc Length here:https://brainly.com/question/32035879
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A builder wishes to construct a ramp 21 feet long that rises to a height of 5 feet above level ground. What angle should the ramp make with the horizontal? Decimal approximations will be marked incorrect. Symbolic trigonometric expressions such as arctan(5) are accepted. 1/ 5 21 α cos radians
Answer:
α = 13.39249775
Step-by-step explanation:
Length of the vamp = 21 ft long
Height of the vamp= 5 ft
using SOHCAHTOA
tanα = opposite /adjacent
Tan α = 5/21
α = arctan(5/21)
α = 13.39249775
Mars Inc. claims that they produce M&Ms with the following distributions:
Brown 30% Red 20% Yellow 20%
Orange 10% Green 10% Blue 10%
How many M&Ms must be sampled to construct the 97% confidence interval for the proportion of red M&Ms in that bag if we want a margin of error of ± .15?
a) 33
b) 36
c) 34
d) 26
e) 25
f) None of the above
Answer:
C) 34
Step-by-step explanation:
1) Some definitions
By definition the margin of error (ME) is the error that tell to us how many percentage points your results will differ from the real population value, on this case our parameter of interest is pr = proportion of red M&M's
ME = Critical value x Standard error of the sample =0.15.
The proportion of red M&M's follows a normal distribution, and our critical value would be from the normal standard distribution on this case
2) Calculate the critical value
a) Compute alpha (α): α = 1 - (confidence level / 100) = 1- 0.97 = 0.03
b) Calculate the critical probability (p*): p* = 1 - α/2 = 1 - (0.03/2) = 0.985
c) Find the z-score using the cumulative probability obtained at step b)
On this case P(Z<z) = 0.985 , the value of z = 2.17 using the normal standard table
3) Calculate n from the formula of ME
The margin of error for a proportion is given by this formula
ME = z sqrt{{pr(1-pr)/n}}
Squaring both sides :
(ME/z) ^2 = (pr(1-pr))/n
And solving for n we got
n = (pr(1-pr))/(ME/z)^2 = (0.2x0.8)/ (0.15/2.17)^2 = 33.488
We need to round up the sample in order to ensure that the confidence level of 97% is meeted, and on this case the answer would be 34.
The number of M&Ms needed to construct a confidence interval for the proportion of red M&Ms with a margin of error of ± .15 is n = 36.
To construct a confidence interval for the proportion of red M&Ms with a margin of error of ± .15, we can use the formula for the sample size needed:
n = (Z² × p × (1-p)) / E²
Substitute Z = 2.17 (for 97% confidence), p = 0.20 (given proportion of red M&Ms), and E = 0.15 into the formula:
n = (2.17²× 0.20 × 0.80) / 0.15²
= 36
Therefore, the number of M&Ms that must be sampled to construct the 97% confidence interval for the proportion of red M&Ms with a margin of error of ± .15 is 36.
find the value(s) of the missing variables
Answer: x = 9
y = 13
Step-by-step explanation:
In the last math test andrew answered 80% of all the questions correctly.If he answered 32 questions correctly, what was the total number of questions in the math test?
Answer:
The total number of questions are 40
Step-by-step explanation:
let the total number of questions be "x".
percent of questions answered correctly are 80.
In fractions 80% is [tex]\frac{4}{5}[/tex]
The total number of correctly answered questions are 32.
the equation is,
[tex](\frac{4}{5})(x) = 32[/tex]
thus , x = [tex]\frac{(5)(32)}{4}[/tex]
x= 40
Which statement best describes a polygon?
A. a closed plane figure with three or more sides that are straight
B. a closed plane figure with four or more sides
C. an open or closed plane figure
D. an open or closed plane figure with three or more sides Reset Next
The best statement that could be used describe a polygon is that it is a closed plane figure with three or more sides that are straight. Thus, the correct answer is A: a closed plane figure with three or more sides that are straight.
Hope this helps! :)