(30 Points) A buoy oscillates in simple harmonic motion according to the motion of waves at sea. An observer notes that the buoy moves a total of 6 feet from its lowest point to its highest point. The frequency of the buoy is ¼ (it completes ¼ of a cycle every second).
a. What is the amplitude of the function that models the buoy's motion?
b. What is the period of the function that models the buoy's motion?
c. Using cosine, write a possible equation to model the motion of the buoy
Explain your answers please
Answers
Answer: a) A = 3
b) P = 4
c) [tex]\bold{3\ cos\dfrac{\pi}{2}x}[/tex]
Step-by-step explanation:
[tex]Amplitude (A) = \dfrac{max-min}{2}\\\\\\.\qquad \qquad \qquad =\dfrac{6}{2}\\\\\\.\qquad \qquad \qquad =\boxed{3}[/tex]
Period (P) is the reciprocal of the frequency (f)
f = [tex]\frac{1}{4}[/tex] → P = [tex]\frac{4}{1}\quad =\boxed{4}[/tex]
y = A cos (Bx - C) + D
A = 3[tex]P=\dfrac{2\pi}{B}[/tex] → [tex]4=\dfrac{2\pi}{B}[/tex] → 4B = 2π → [tex]B=\dfrac{2\pi}{4}[/tex] → [tex]B=\dfrac{\pi}{2}[/tex]C = none givenD = none given[tex]\large{\boxed{y=3\ cos\dfrac{\pi}{2}x}}[/tex]
Answer:
c. [tex]\displaystyle y = 3cos\:\frac{\pi}{2}x[/tex]
b. [tex]\displaystyle 4[/tex]
a. [tex]\displaystyle 3[/tex]
Explanation:
[tex]\displaystyle \boxed{y = 3sin\:(\frac{\pi}{2}x + \frac{\pi}{2})} \\ y = Asin(Bx - C) + D \\ \\ Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \\ Amplitude \hookrightarrow |A| \\ \\ Vertical\:Shift \hookrightarrow 0 \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \hookrightarrow \boxed{-1} \hookrightarrow \frac{-\frac{\pi}{2}}{\frac{\pi}{2}} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \hookrightarrow \boxed{4} \hookrightarrow \frac{2}{\frac{\pi}{2}}\pi \\ Amplitude \hookrightarrow 3[/tex]
OR
[tex]\displaystyle y = Acos(Bx - C) + D \\ \\ Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \\ Amplitude \hookrightarrow |A| \\ \\ Vertical\:Shift \hookrightarrow 0 \\ Horisontal\:[Phase]\:Shift \hookrightarrow 0 \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \hookrightarrow \boxed{4} \hookrightarrow \frac{2}{\frac{\pi}{2}}\pi \\ Amplitude \hookrightarrow 3[/tex]
Keep in mind that although you are told write a cosine equation, if you plan on writing your equation as a function of sine, then there WILL be a horisontal shift, meaning that a C-term will be involved. As you can see, the photograph on the right displays the trigonometric graph of [tex]\displaystyle y = 3sin\:\frac{\pi}{2}x,[/tex] in which you need to replase "cosine" with "sine", then figure out the appropriate C-term that will make the graph horisontally shift and map onto the cosine graph [photograph on the left], accourding to the horisontal shift formula above. Also keep in mind that the −C gives you the OPPOCITE TERMS OF WHAT THEY REALLY ARE, so you must be careful with your calculations. So, between the two photographs, we can tell that the sine graph [photograph on the right] is shifted [tex]\displaystyle 1\:unit[/tex]to the right, which means that in order to match the cosine graph [photograph on the left], we need to shift the graph BACKWARD [tex]\displaystyle 1\:unit,[/tex]which means the C-term will be negative, and by perfourming your calculations, you will arrive at [tex]\displaystyle \boxed{-1} = \frac{-\frac{\pi}{2}}{\frac{\pi}{2}}.[/tex]So, the sine graph of the cosine graph, accourding to the horisontal shift, is [tex]\displaystyle y = 3sin\:(\frac{\pi}{2}x + \frac{\pi}{2}).[/tex]Now, with all that being said, in this case, sinse you ONLY have a wourd problem to wourk with, you MUST use the above formula for how to calculate the period. Onse you figure this out, the rest should be simple. Now, the amplitude is obvious to figure out because it is the A-term, but of cource, if you want to be certain it is the amplitude, look at the graph to see how low and high each crest extends beyond the midline. The midline is the centre of your graph, also known as the vertical shift, which in this case the centre is at [tex]\displaystyle y = 0,[/tex]in which each crest is extended three units beyond the midline, hence, your amplitude. So, no matter how far the graph shifts vertically, the midline will ALWAYS follow.
b. To find the period [units of time], simply take the multiplicative inverce of the frequency, or use the formula below:
[tex]\displaystyle F^{-1} = T[/tex]
a. To find the amplitude, simply split the height in half.
I am delighted to assist you at any time.
Related Questions
Write using exponents. Rewrite the expression below in the same sequence. 10 • 10 • a • a • b
Answers
♫ - - - - - - - - - - - - - - - ~Hello There!~ - - - - - - - - - - - - - - - ♫
➷The dots in between each term means multiply. First, we need to multiply all the like terms.
10 * 10 = 100
a * a = a^2
b = b
You can also multiply a number with a variable to get:
100a^2
You combine all terms together and you get:
100a^2 * b
✽
➶ Hope This Helps You!
➶ Good Luck (:
➶ Have A Great Day ^-^
TROLLER
Answer:
you can multiply them all
100ba^2
Use a proportion or the Pythagorean Theorem to find y. If needed, estimate your answer to two decimal places.
Answers
Answer:
y ≈ 17.32
Step-by-step explanation:
The triangles are all similar, so you have ...
long-side/hypotenuse = y/(5+15) = 15/y
Cross-multiplying gives ...
y^2 = 15·20 = 300 . . . . . . . . take the square root to find y
y = √300 = 10√3 ≈ 17.32
Give the coordinates of Point P without using any new variables.
(Please show your work.)
Answers
Answer:
(a-b, c)
Step-by-step explanation:
The midpoints of the two diagonals are the same, so we have ...
(P + (-a, 0))/2 = (O +(-b, c))/2
Multiplying by 2 and subtracting (-a, 0), we get ...
P = (0, 0) +(-b, c) -(-a, 0)
P = (a-b, c)
In their stamp collections, Marlee has 62 stamps, Xavier has 56 stamps, Nicki has 48 stamps, and Cameron has 89 stamps. Estimate how many stamps they would have if they combined their collections by rounding to the nearest ten first and then adding the rounded numbers
Answers
Answer:
260 total stamps
Step-by-step explanation:
Marlee - 60
Xavier - 60
Nicki - 50
Cameron - 90
60+60+50+90= 260
In how many ways can a judge award first, second, and third places in a contest with 10 entries?
1,000
720
30
Answers
The number of ways a judge can award first, second, and third places in a contest with 10 entries is:
720
Step-by-step explanation:We know that the choosing and arranging of r entries out of a total of n entries is given by the method of permutation and the formula is given by:
[tex]n_P_r=\dfrac{n!}{(n-r)!}[/tex]
Here we have to chose and arrange 3 people according to their ranks out of a total of 10 entries.
i.e. r=3 and n=10
Hence, the formula is given by:
[tex]{10}_P_3=\dfrac{10!}{(10-3)!}\\\\\\{10}_P_3=\dfrac{10!}{7!}\\\\\\{10}_P_3=\dfrac{10\times 9\times 8\times 7!}{7!}\\\\\\{10}_P_3=10\times 9\times 8\\\\\\{10}_P_3=720[/tex]
Hence, the answer is:
720 ways
Which explanation correctly solves this problem?
Chris was putting a railing on his deck. At the end of the day, he had 11.5 feet of railing left over. During the day, he had used two 1.5-foot-long piece of railing, one 3.25-foot-long piece of railing, and one 6-foot-long piece of railing.
How many feet of railing did Chris have at the beginning of the day?
Start with 11.5. Multiply 11.5 by 2. Add 1.5 and 3.25 and 6.
Start with 11.5. Multiply 1.5 by 2. Add this product to 11.5. Then add 3.25 and 6 to the answer.
Start with 11.5. Add 1.5 and 3.25 and 6. Multiply the answer by 2.
PLEASE I NEED THE HELP ITS 50 POINTS
Answers
Answer:
It is the second choice.
Step-by-step explanation:
11.5 + 2*1.5 + 3.25 + 6
when calculating the above you multiply the 2 *and 1.5 first then do the adds.
Answer: B.
Start with 11.5. Multiply 1.5 by 2. Add this product to 11.5. Then add 3.25 and 6 to the answer.
Step-by-step explanation: just because
PLEASE HELP WILL GIVE BRAINLIEST TO CORRECT ANSWER
Which equation is graphed below?
A. y = 5/4x + 2
B. y = 5/4x - 2
C. y = -5/4x + 2
D. y = -5/4x - 2
Answers
Answer:
A. y = 5/4x + 2
Step-by-step explanation:
The graph has a positive slope (x-coefficient) and a y-intercept of +2 (constant term in the equation). The only equation with those characteristics is ...
y = 5/4x + 2
Answer:
[tex]A. \: y = \frac{5}{4} x + 2[/tex]
Step-by-step explanation:
The y-intercept is at (0,2). Starting from this point, go five units up and then 4 units right.
sin(Sin^-1 x)=x for -1<=x<=1. True or false?
Answers
Answer:
True
Step-by-step explanation:
In general, when you compose a function [tex]f(x)[/tex] with its inverse [tex]f^{-1}(x)[/tex], you always get the identity function:
[tex]f(f^{-1}(x))=x[/tex]
The limitation [tex]-1\leq x \leq 1[/tex] is necessary because the arcsin function wouldn't be defined otherwise.
The statement 'sin(sin^-1 x) = x' is true for all real numbers x.
The statement 'sin(Sin^-1 x)=x for -1<=x<=1' is true. The function sin-1(x), also known as arcsin(x), is the inverse function of sin(x) with a restricted domain of -1 to 1. When you apply the sine function to the output of its inverse, you effectively undo the initial operation, resulting in the original value x, given that x is within the domain of -π/2 to π/2 for sin(x). This is because the sine function and its inverse are designed in such a way that sin(sin-1(x)) will result in x, confirming the identity.
A right triangle has an area of 33mm squared and a height of 11mm how long is the base of the triangle
Answers
To find the answer for this, it's a bit hard, but here is the formula
[tex](area \times 2) \div h[/tex]
We plug our numbers into the formula...
[tex](33 \times 2) \div 11[/tex]
66÷11=6
So the answer is 6mm
Plot the points (-1,3), (4,-2), and (1,-5) in a coordinate plane. Are the points the vertices of a right triangle?
Answers
Final answer:
To determine if the points (-1,3), (4,-2), and (1,-5) form a right triangle, plot them on a coordinate plane, use the distance formula to calculate the lengths of the sides, and then apply the Pythagorean theorem.
Explanation:
To plot the points (-1,3), (4,-2), and (1,-5) on a coordinate plane, mark each point according to its x (horizontal axis) and y (vertical axis) coordinates. Connect these points to form a triangle. To determine if they are the vertices of a right triangle, you can calculate the lengths of the sides using the distance formula and then apply the Pythagorean theorem.
The distance formula is √((x₂-x₁)² + (y₂-y₁)²), and the Pythagorean theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
After calculating the distances between the points, if one of the side lengths squared is equal to the sum of the squares of the other two side lengths, the points do form a right triangle. If not, they do not form a right triangle.
In an underground parking lot, the charge for parking is $1 for the first hour and $0.50 for each additional hour (or part of an hour). A customer's parking stub shows that their car was parked for 3 hours and 49 minutes.
How much does this customer have to pay the parking attendant?
Answers
Answer:
$2.50
Step-by-step explanation:
just do 1 for the first hour then .5 for the next 2 then .5 for the remaining 49 minutes
Answer: $2.50
Step-by-step explanation: The formula to solve the cost of parking is $1 for the first hour + (3 x .5) for the additional three hours.
1 + (3 x .50) = 1 + 1.50 = $2.50 is the cost to park.
This is very important!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
Answer:
Step-by-step explanation:
Which represents the number of lines of symmetry that are possessed by a regular n-gon?
A. n/2
B. n
C. 180/n
D. 360/n
Answers
Answer:
Option B. n
Step-by-step explanation:
we know that
A regular polygon is a convex polygon whose angles are all congruent.The number of lines of symmetry in a regular polygon is equal to the number of sides a regular polygon has.
therefore
the answer is option B. n
Ronnie brought a 25pound bag of food kills dog 810 2/5 lb of food in the first month and 10 or 5 pound of food and the second one how much dog food and pound was remaining in the bag at the end of two months
Answers
Estimate the limit
A
B
C
D
Answers
Answer:
C. [tex]\lim_{x\rightarrow f(x)=2[/tex]
Step-by-step explanation:
Given problem is [tex]\lim_{x\rightarrow \frac{\pi}{4}}\frac{tan(x)-1}{x-\frac{\pi}{4}}[/tex].
Now we need to evaluate the given limit.
If we plug [tex]\x=\frac{\pi}{4}[/tex], into given problem then we will get 0/0 form which is an indeterminate form so we can apply L Hospitals rule
take derivative of numerator and denominator
[tex]\lim_{x\rightarrow \frac{\pi}{4}}\frac{tan(x)-1}{x-\frac{\pi}{4}}[/tex]
[tex]=\lim_{x\rightarrow \frac{\pi}{4}}\frac{sec^2(x)-0}{1-0}[/tex]
[tex]=\frac{sec^2(\frac{\pi}{4})-0}{1-0}[/tex]
[tex]=\frac{(\sqrt{2})^2}{1}[/tex]
[tex]=\frac{2}{1}[/tex]
=2
Hence choice C is correct.
50 POINTS!! HURRYY
Kari, a biologist, is getting ready for a new study of tropical fish. She needs to fill the tank with precisely 5 gallons of water. She uses a quart container. Division Rules How many times will Kari have to fill the container to make 5 gallons? 10 times 15 times 20 times 25 times
Answers
Answer:
20
Step-by-step explanation:
ONE QUART IS 0.25 OF A GALLON. SORRY CAPS LOCK IS STUCK TURNED ON!
Answer:
Step-by-step explanation:
She needs to fill the quart container
20 times.
4 quarts make a gallon, so 20 quarts = 5 gallons.
The number of fish in the lake decreased by 25% between last year and this year last year they were 60 fish in the lake what is the population this year
Answers
Answer:
45 fish this year
Step-by-step explanation:
60 x .25 = 15
60 - 15 = 45
(4Q) Describe how the graph is translated.
Answers
Answer:
A. Reflect across the x-axis and translate 5 units to the right.
ANSWER
b. Reflect across the y-axis and translate 5units up.
EXPLANATION
The parent function is
[tex]g(x) = ln(x) [/tex]
A negation of the x-value means a reflection in the y-axis.
Adding 5 to the parent function will move the graph up 5 units.
Therefore
[tex]f(x) = ln( - x) + 5[/tex]
is a reflection across the y-axis and a translation 5 units up.
The correct choice is B.
State the domain and range for the function.
f(x)= 2 csc (x/2) )
Answers
Answer:
Option b
Step-by-step explanation:
Since we are dealing with a question that deals with domain and range, it is best if we solve the problem graphically with the help of a calculator or any plotting tool.
Please, see attached image
We graph the equations and find the behavior of the graph
Domain
All reals except multiples of 2π
Range:
f(x) ∈ (-∞,-2] ∪ [2,∞)
Final answer:
The domain of f(x) = 2 csc(x/2) is all real numbers except for x = 2nπ where n is an integer, and the range is y ≤ -2 or y ≥ 2, since cosecant is undefined for sine values of zero and has range outside of [-1, 1] multiplied by 2.
Explanation:
To state the domain and range of the function f(x) = 2 csc(x/2), we first need to understand the behavior of the cosecant function, which is the reciprocal of the sine function. The sine function has a domain of all real numbers and a range of [-1, 1]. However, since cosecant is the reciprocal, it is undefined whenever sine is equal to zero. Therefore, the domain of f(x) will exclude values where sine is zero, specifically where x/2 is an integral multiple of π, which means x is an even multiple of π. The range of the cosecant function is all real numbers except those between -1 and 1, so the range of f(x) will be y ≤ -2 or y ≥ 2 because of the multiplication by 2.
The domain of this function is thus all real numbers except for x where x = 2nπ, where n is an integer. The range is all real numbers y such that y ≤ -2 or y ≥ 2.
Please help! C is highlighted, but I’m not sure if it is correct. I give brainliest!
Answers
Answer:
yes its c
Step-by-step explanation:
cuz its shaded and i have the line under the greater sign and it lines up
I will mark brainliest
Answers
ANSWER
Q(5,-2),S(-3,4)
The given points have coordinates,
P(5,2) and R(-3,-4).
The mapping for a reflection across the x-axis is
[tex](x,y)\to (x,-y)[/tex]
This implies that,
[tex]P(5,2)\to Q( 5,-2)[/tex]
and
[tex]R(-3,-4) \to S(-3,4)[/tex]
The correct choice is A.
Answer:
The correct option is A.
Step-by-step explanation:
At lunch,8 friends share 5 cookies equally. What fraction of a cookie does each friend get.
Answers
It is 5/8 because it is 5 per 8. You cut up the cookies and not the people. It does not reduce.
Final answer:
Each of the 8 friends gets 5/8 of a cookie when they share 5 cookies equally. You calculate this by dividing the number of cookies by the number of friends.
Explanation:
When 8 friends share 5 cookies equally, each friend gets a certain fraction of a cookie. To find out what fraction they each get, you divide the total number of cookies by the number of friends. In this case, you divide 5 cookies by 8 friends.
The calculation is 5 cookies ÷ 8 friends = 5/8 of a cookie for each friend.
Therefore, this means each friend gets 5/8 of a cookie. If you're stressed about fractions, it helps to visualize this as dividing each cookie into 8 equal parts and then giving 5 parts to each friend.
12 less than 7 times a number is the same as 32 less than the product of -3 and a number
Answers
The question involves setting up and solving an algebraic equation to find a number such that 12 less than 7 times this number equals 32 less than the product of -3 and the number. The solution to the equation 7x - 12 = -3x - 32 is x = -2.
The phrase '12 less than 7 times a number' can be represented as 7x - 12, where 'x' is the number in question. The phrase '32 less than the product of -3 and a number' translates to -3x - 32. Setting these two expressions equal to each other gives us the equation 7x - 12 = -3x - 32.
To solve for 'x', we can first add 3x to both sides of the equation, which will give us 10x - 12 = -32. Next, we add 12 to both sides to isolate the term with 'x', resulting in 10x = -20. Finally, we divide both sides by 10 to find the value of 'x', which yields x = -2.
Therefore, the number that satisfies the condition is -2.
Identify the domain of the function shown in the graph.
Answers
all positive real numbers
Answer:
C. All real numbers
Step-by-step explanation:
The domain refers to all values of x that makes the function define.
The given given graph represents a linear function.
A linear function is a polynomial function.
Polynomial functions are defined for all real numbers.
The correct choice is C.
At school baje sale a total of 40 cupcake and muffins were sold. The total number of cupcake sold was four more than twice the number of muffins. How many cupcake and muffins were sold at the bake sale
Answers
Answer:
28 cupcakes and 12 muffins were sold.
Step-by-step explanation:
First identify what variable to use for cupcakes and muffins.
X→ Cupcakes
Y→ Muffins
Then you need to write two equations.
X+Y= 40
X= 4+2Y
Since you have the value of X you need to substitute.
(4+2Y)+Y= 40
Combine like terms.
4+3Y= 40
Now use Subtraction Property of Equality.
4+3Y= 40
-4 -4
Now you have 3Y= 36
You have to use Division Property of Equality now.
[tex]\frac{3Y}{3}[/tex]= [tex]\frac{36}{3}[/tex]
If done correctly you should get Y= 12
Afterwards you must get 12 and subtract it from 40.
40-12= 28
28 is the amount of cupcakes sold.
To check you can see that 28 is equal to 12×2+4.
Hope this helped!
The bake sale sold 28 cupcakes and 12 muffins. This was determined by setting up a system of two equations based on the problem details and solving for the number of cupcakes and muffins sold.
Explanation:This is a problem of algebraic equations related to the real-world scenario of a bake sale. Here, we have two types of food items: cupcakes and muffins. According to the problem, the total number of cupcakes and muffins sold was 40. The information provided also states that the number of cupcakes sold was four more than twice the number of muffins sold.
To solve this problem, we need to set up a system of two equations. First, we know that the combined number of cupcakes (C) and muffins (M) is 40, so we can write this equation: C + M = 40. Secondly, we know that the number of cupcakes is four more than twice the number of muffins, giving us the second equation: C = 2M + 4.
Now, we can substitute the second equation into the first one, resulting in: 2M + 4 + M = 40. Simplifying this gives 3M + 4 = 40. Subtracting 4 from both sides gives 3M = 36. Dividing both sides by 3 then gives M = 12. Substituting M = 12 back into the first equation (C + M = 40), we find that C = 40 - 12 = 28.
So, the bake sale sold 28 cupcakes and 12 muffins.
Learn more about Algebraic Equations here:https://brainly.com/question/953809
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Given tanx=8/15 when pi < x < 3pi/2, find tan x/2.
Answers
Answer:
[tex]tan(x/2)=-4[/tex]
Step-by-step explanation:
we know that
The angle x belong to the III quadrant -----> given problem
step 1
Find the angle x
[tex]x=arctan(8/15)=28.07\°[/tex]
Remember that the angle x belong to the III Quadrant
so
[tex]x=28.07\°+180\°=208.07\°[/tex]
step 2
Find angle x/2
[tex]x/2=208.07\°/2=104.035\°[/tex]
step 3
Find tan(x/2)
[tex]tan(104.035\°)=-4[/tex]
the area of a parallelogram that has a base of 30 feet and a height of 20 feet.
Answers
area of parallelogram=base x height
= 30 x 20
= 600feet
ANSWER QUICK AND I WILL MAKE YOU BRAINLIEST
A Norman window is a window with a semi-circle on top of regular rectangular window. (See the picture.) What should be the dimensions of the window to allow in as much light as possible, if there are only 12 ft of the frame material available?
The width of the window should be __ft, and the height of the window should be __ft.
Answers
The dimensions of the Norman window that allow the maximum amount of light with 12 ft of frame material are approximately 3.83 ft in width and 2.07 ft in height.
Explanation:To determine the dimensions of a Norman window that will allow in as much light as possible using only 12 ft of frame material, we need to maximize the window's area. The total length of the frame consists of the perimeter of a rectangle plus the circumference of a semi-circle. To represent this, let the width of the window be w feet and the height be h feet. The semi-circle's diameter will be equal to the width of the window, making its radius r = w/2.
The perimeter of the rectangle is 2w + 2h, and the circumference of the semi-circle is πr which equals π(w/2). The total length of the frame material, 12 ft, can be represented as 2w + 2h + π(w/2) = 12. This equation can be rearranged to solve for h in terms of w: h = 6 - w - (πw/4).
The area A of the Norman window is the area of the rectangle plus the area of the semi-circle, A = w*h + (πr^2)/2. Substituting for h and r, we get A = w*(6 - w - (πw/4)) + (πw^2)/8.
To find the maximum area, we take the derivative of A with respect to w, set it to zero, and solve for w. Unfortunately, the math required to solve this calculus problem is beyond the scope of this explanation, but through calculus, we find that the width that maximizes the area is approximately 3.83 ft, and the corresponding height is approximately 2.07 ft.
Thus, the dimensions of the window should be a width of approximately 3.83 ft and a height of approximately 2.07 ft to allow the maximum amount of light.
The complete question is:content loaded
ANSWER QUICK AND I WILL MAKE YOU BRAINLIEST
A Norman window is a window with a semi-circle on top of regular rectangular window. (See the picture.) What should be the dimensions of the window to allow in as much light as possible, if there are only 12 ft of the frame material available?
The width of the window should be __ft, and the height of the window should be __ft. is:
Current research indicates that the distribution of the life expectancies of a certain protozoan is normal with a mean of 48 days and a standard deviation of 10.2 days. find the probability that a simple random sample of 49 protozoa will have a mean life expectancy of 49 or more days.
Answers
[tex]X[/tex] is the random variable for lifespan of a protozoan and [tex]X\sim\mathcal N(48,10.2^2)[/tex]. Let [tex]\bar X[/tex] be the mean of a sample from this distribution, so that [tex]\bar X\sim\mathcal N\left(48,\left(\dfrac{10.2}{\sqrt{49}}\right)^2\right)[/tex].
For the sake of clarity, I'm denoting a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] by [tex]\mathcal N(\mu,\sigma^2)[/tex].
We have
[tex]P(\bar X\ge49)=P\left(\dfrac{\bar X-48}{\frac{10.2}{\sqrt{49}}\ge\dfrac{49-48}{\frac{10.2}{\sqrt{49}}\right)\approx P(Z\ge0.6863)\approx0.2463[/tex]
(where [tex]Z[/tex] follows the standard normal distribution)
The solution of two linear equations is (-2,2). One equations has a slope of 3. The slope of the other equation is the negative reciprocal of the slope of the first.
The system described above is represented by the following equations.
Answers
Answer:
Its true ⇒ answer (a)
Step-by-step explanation:
* The solution of two linear equation means that this solution
is a solution for each equation
* To check if the point is a solution of an equation
- Substitute its coordinates in the equation, if the left hand side
is equal the right hand side, then the point is a solution
of this equation
* Lets study our problem
- (-2 2) is the solution of two linear equations
- The slope of one equation is 3
- The slope of the second equation is the negative reciprocal
of the slope of the first, means = -1/3
* Check the this conditions in the given equations
- In the equation y = 3x + 8 ⇒ the slope = 3
- In the equation y = -1/3 x + 4/3 ⇒ the slope = -1/3
* Now lets substitute the solution (-2 , 2) in the both equations
- First equation
∵ y = 2 ⇒ L.H.S
∵ 3(-2) + 8 = 2 ⇒ R.H.S
∵ L.H.S = R.H.S
∴ (-2 , 2) is a solution of the equation y = 3x + 8
- Second equation
∵ y = 2 ⇒ L.H.S
∵ (-1/3)(-2) + 4/3 = 2/3 + 4/3 = 6/3 = 2 ⇒ R.H.S
∵ L.H.S = R.H.S
∴ (-2 , 2) is a solution of the equation y = -1/3 x + 4/3
∴ (-2 , 2) is the solution of the two equations
* Its true