Answer: [tex]\Large\boxed{\dfrac{3}{8} }[/tex]
Step-by-step explanation:
Given information
[tex]b=2[/tex]
[tex]c=-\dfrac{7}{4}[/tex]
Given expression
[tex]b-(-\dfrac{1}{8}) +c[/tex]
Substitute values into the expression
[tex]=(2)-(-\dfrac{1}{8}) +(-\dfrac{7}{4} )[/tex]
Convert the fractions into the common denominator
Least Common Multiple (LCM) of 8 and 4 = 8
[tex]=(2)-(-\dfrac{1}{8}) +(-\dfrac{7\times2}{4\times2} )[/tex]
[tex]=(2)-(-\dfrac{1}{8}) +(-\dfrac{14}{8} )[/tex]
Simplify the parenthesis
[tex]=2+\dfrac{1}{8} -\dfrac{14}{8}[/tex]
Simplify by addition
[tex]=\dfrac{16}{8} +\dfrac{1}{8} -\dfrac{14}{8}[/tex]
[tex]=\dfrac{17}{8} -\dfrac{14}{8}[/tex]
Simplify by subtraction
[tex]\Large\boxed{=\dfrac{3}{8} }[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Find the value of u -8 given that -7u-5=2
Answer: u - 8 = -9
Step-by-step explanation:
Given equation
-7u - 5 = 2
Add 5 on both sides
-7u - 5 + 5 = 2 + 5
-7u = 7
Divide -7 on both sides
-7u / (-7) = 7 / (-7)
u = -1
Substitute values into the goal expression
u - 8 = (-1) - 8 = [tex]\Large\boxed{-9}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
The answer is -9.
First, let's find the value of u.
-7u - 5 = 2-7u = 7u = -1Now, substitute in the expression.
u - 8(-1) - 8-9Instructions: Match the following data with the correct stem and leaf.
Helped need please
Based on the stem and leaf plots given, the correct stem and leaf plot for the first table on per capita income is the third stem and leaf diagram.
For the table on the basketball tornament, the correct stem and leaf plot would be the last one.
How are stem and leaf plots used?First look at the key to the bottom right of the plot to see how to interpret the data.
The key on the third stem and lead plot says 1 I 2 = 12,000. This means that every steam and lead pairing is in thousands.
Based on the income per capita being in thousands, this is the most likely stem and lead plot for the first table on income per capita.
The second table about basketball tornament appearances is on single and double digits. Only a small stem and leaf key will do.
Look at the number of appearances. The minimum is 1. On a stem and leaf, this is shown as 0 I 1. Options A and B don't have this so we go to Options C and D.
As established, Option C is for data in their thousands so it is out. The correct stem and leaf plot is therefore Option D.
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A man gave out $24000 to his three brothers A,B and C to be shared among themselves . If A takes twice as much as B and B is given one third of what C takes , how much did each of them receive?
Answer:
A = 8000
B = 4000
C = 12000
Step-by-step explanation:
$24000 = A,B and C
A = 2B
B = 1/3C
solve for the letter that gets described the most
that letter is B
A =2B
B = 1/3C -> C = 3B
24000 = A + B + C
24000 = 2B + B + 3B
24000 = 6B
B = 4000
A =2B = 8000
C = 3B = 12000
Sims waist is 30 inches. He wants to put on more weight and is hoping to gain enough weight to increase his waist size by 8%. How many inches does he want his waist to be?
Sim wants to increase his waist size to 32.4 inches.
How many inches does Sim want his waist to be?Sim wants the increase to be 8% of 30 inches.
The amount of increase Sim wants in his waist size exists
8% × 30 inches.
= 0.08 × 30 inches.
= 2.4 inches.
Adding the wanted increase to his present size would create Sim's waist size to be
30 in + 2.4 in = 32.4 inches.
Sim wants to increase his waist size to 32.4 inches.
Therefore, the correct answer is 32.4 inches.
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A scientist is tagging fish in the Allentown Lake. She catches a fish, tags it,and releases it back into the water. The scientist knows the exact number of fish in the lake: 32 bass,14 fluke,10 carp, and 3 bull sharks.
Calculate the probabilities that the scientist will catch:
A. a bass and then a carp
B. two consecutive bull sharks
C. a fluke and then any different type of fish
d. a bass, another bass, and a bull shark
Probability has to do with the possibility that an event may or may not happen.
What is the probability?The term probability has to do with the possibility that an event may or may not happen. The value of probability always ranges from zero to one.
Now;
Total number of fishes = 32 + 14 + 10 + 3 = 59
a) probability of catching a bass and then a carp = 32/59 * 10/59 = 320/3481
b) probability of catching two consecutive bull sharks= 3/59 * 3/59 = 9/3481
c) probability of catching a fluke and then any different type of fish 14/59 * (32/59 + 10/59 + 3/59) = 14/59 * 45/59 = 630/3481
d) probability of catching a bass, another bass, and a bull shark = 32/59 * 32/59 * 3/59 = 3072/205379
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Sketch the graphic y=|x+1|
Answer:
Consider the table for y= |x+1| :
x | y
---------
0 1
1 2
2 3
-1 2
-2 3
This would give us the parent function of y=|x| but translated up one unit. It should look like a v starting at (0, 1)
chef John bought 3 dozen eggs for $16.20 what is the unit price a $10.80 per 2 dozen eggs b $5.40 per 12 eggs c $0.45 per eggs d $0.22 per eggs e none of these f I don't know this yet
Answer:
0.45 per egg
Step-by-step explanation:
since it's 3 dozen that mean it's gonna be 36 eggs divided by 16.20 so it's 0.45 per egg
Is
7 3
a proper, an improper, a mixed number, or a decimal fraction?
a proper fraction
an improper fraction
a mixed number
a decimal number
Answer: an improper fraction
Step-by-step explanation: An improper fraction is a fraction that is larger than 1 7/3 is larger than one because 7, the numerator is larger than the denominator so 7/3 is an improper fraction
2X/x+2 = - (6/x+4)
This is a problem which I think I’m supposed to get a form like x^2+x+1
Step-by-step explanation:
it's 2x²+8x
and I cannot understand your question
Answer:
x = - 6, x = - 1
Step-by-step explanation:
[tex]\frac{2x}{x+2}[/tex] = - [tex]\frac{6}{x+4}[/tex] ( cross- multiply )
2x(x + 4) = - 6(x + 2) ← distribute parenthesis on both sides )
2x² + 8x = - 6x - 12 ( subtract - 6x - 12 from both sides )
2x² + 14x + 12 = 0 ( divide through by 2 )
x² + 7x + 6 = 0 ← in standard form
(x + 6)(x + 1) = 0 ← in factored form
equate each factor to zero and solve for x
x + 6 = 0 ⇒ x = - 6
x + 1 = 0 ⇒ x = - 1
Target Costing
Instant Image Inc. manufactures color laser printers. Model J20 presently sells for $460 and has a product cost of $230, as follows:
Direct materials $175
Direct labor 40
Factory overhead 15
Total $230
It is estimated that the competitive selling price for color laser printers of this type will drop to $400 next year. Instant Image has established a target cost to maintain its historical markup percentage on product cost. Engineers have provided the following cost-reduction ideas:
Purchase a plastic printer cover with snap-on assembly, rather than with screws. This will reduce the amount of direct labor by 15 minutes per unit.
Add an inspection step that will add six minutes per unit of direct labor but reduce the materials cost by $20 per unit.
Decrease the cycle time of the injection molding machine from four minutes to three minutes per part. Forty percent of the direct labor and 48% of the factory overhead are related to running injection molding machines.
The direct labor rate is $30 per hour.
a. Determine the target cost for Model J20, assuming that the historical markup on product cost and selling price are maintained.
$fill in the blank 1
per unit
b. Determine the required cost reduction.
$fill in the blank 2
per unit
c. Evaluate the three engineering improvements together to determine if the required cost reduction (drift) can be achieved. Do not round interim calculations but round your final answers to two decimal places.
1. Direct labor reduction $fill in the blank 3
per unit
2. Additional inspection $fill in the blank 4
per unit
3. Injection molding productivity improvement $fill in the blank 5
per unit
Total savings $fill in the blank 6
per unit
a) The target cost for Model J20, assuming that the historical markup on product cost and selling price is maintained, is $200 per unit.
b) The required cost reduction per unit is $30 per unit ($230 - $200).
c) An evaluation of the three engineering improvements together to determine if the required cost reduction (drift) can be achieved as follows:
While the calculated cost reduction, including direct materials, is $25, the required reduction to achieve the target markup cost is $30.
Thus, the expected target cost reduction does not add up with the calculated cost reduction, envisaged by the engineers.
Data and Calculations:Model J20 Selling price per unit = $460
Direct materials $175
Direct labor 40
Factory overhead 15
Total $230
Direct labor rate = $30 per hour
Direct labor rate per minute = $0.50 ($30/60)
Direct labor hours per unit = 1¹/₃ hours ($40/$30) or 80 minutes
Direct labor hours per unit reduction:Normal direct labor time = 80 minutes
Plastic printer cover = -15 minutes
Total new direct labor time = 65 minutes
Direct labor rate per unit = $32.50 ($0.5 x 65)
Additional inspection time = 6 minutes
Injection molding machine = 1 minute
Direct materials cost = $155 ($175 - $20)
Historical markup = 200% ($460/$230 x 100)
Target cost based on historical markup = $200 ($400 x 100/200)
Required cost reduction per unit is $30 per unit ($230 - $200)
1. Direct labor reduction of $7.50 per unit ($40 - $32.50)
2. Additional inspection $3 per unit (6 x $0.50)
3. Injection molding productivity improvement $0.50 per unit
4. Material cost reduction = $20
Total savings are $25 per unit ($7.50 - $3 + $0.50 + $20).
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There is a boardwalk game at Point Pleasant where you are blindfolded to throw darts at a board full of balloons. Each time a dart is popped, it is not replaced until the next turn. The board has 10 green,4 purple,5 red,and 2 tiedye and 3 black balloons.
Find the probabilities of the following outcomes:
a. Popping two reds consecutively during one turn
b. Popping a red, then a green during one turn
c. Popping a red, then a black, then a red during one turn
d. Popping anything but a tiedyed balloon on three consecutive throws
The given number of balloons, green = 10, purple = 4, red = 5, tie-dye = 2, black = 3, gives;
a. 5/138
b. 25/276
c. 5/1012
d. 35/46
How can the different probabilities be calculated mathematically?Given parameters;
Number of balls;
Green = 10
Purple = 4
Red = 5
Tie-dye = 2
Black = 3
Mode of selection = Without replacement
Number of balloons = 10+4+5+2+3 = 24
a. Probability of popping a red balloon = 5/24
Probability of popping a second red balloon = 4/23
Therefore;
Probability of popping two reds consecutively = 5/24 × 4/23 = 5/138
b. Probability of popping a red balloon = 5/24
Probability of popping a green balloon next = 10/23
Therefore;
Probability of popping a red and then a green balloon = 5/24 × 10/23 = 25/276
c. Probability that the first balloon that pops is a red = 5/24
Next balloon is a black = 3/23
Third balloon is red = 4/22
The probability, P, is therefore;
P = 5/24 × 3/23 × 4/22 = 5/1012d. The probability that the first balloon is a tie-dye = 2/24 = 1/12
Therefore;
Probability that the first balloon is not a tie-dye = 1 - 1/12 = 11/12
Probability that the second balloon is not a tie-dye = 21/23
Similarly;
Probability that the third balloon is not a tie-dye = 20/22 = 10/11
Which gives;
The probability, P, of popping anything but a tie-dye on three consecutive throws is therefore;
P = 11/12 × 21/23 × 10/11 = 35/46Learn more about probability theory in mathematics?
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PLEASE HELP ALOT OF POINTSThe Pythagorean Theorem states "The square on the hypotenuse of a
right triangle is equal to the sum of the squares on the two legs". Jaylene
produces this image in her notebook and claims that it proves the
Pythagorean Theorem is true. Is this a fair claim? Why or why not?
Answer:
choice C. Yes, this image proves the pythagorean theorem is true because 9 + 16 = 25
Step-by-step explanation:
If a certain number, , is multiplied by 6, it will be equal to 4 less than thrice the same number. The correct equation for the above statement is?
The correct equation is [tex]x\cdot6=3x-4[/tex].
Let the number be [tex]x[/tex].
According to the question,
Number multiplied by [tex]6[/tex] [tex]=6x[/tex].
[tex]4[/tex] less than thrice the number [tex]=3x-4[/tex].
Hence, the correct equation will be [tex]x\cdot6=3x-4[/tex].
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Find the upper quartile summary for the data. {51, 49, 52, 46, 50, 38, 38, 45, 34, 52, 46}
The upper quartile for the given data is 51 .
Given data: {51,49,52,46,50,38,38,45,34,52,46}
Data in arranged form: {34,38,38,45,46,46,49,50,51,52,52}
Find the median first, then the upper quartile. There are eleven data points, thus look at term six to determine the median as there are five data points on any side. The median is 46 because the sixth term is 46.
The upper extreme, 52, and the median are then used to determine the upper quartile; make sure not to include the median data point when dividing the groups. The upper quartile is the third term (between the median and upper extreme) because there would be two data points on each side if there were only five data points from the median to the upper extreme. The third term, which is 51, falls into the upper quartile.
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compare the discounts you would receive with a 10% off coupon versus a 510 off coupon. Which one is better? Are there situations in which the other one is better? Will they get you the same amount of discount? Show All the work.
1. To determine the better option between a 10% off coupon versus a $10 off coupon, it is necessary to determine the coupon price.
2. On the other hand, the $10 off coupon becomes better when the coupon price is less than $100.
3. The 10% off coupon and the $10 off coupon do not give the same amount of discount unless the coupon or list price is $100 in both situations.
What is a discount?A discount is a monetary reduction in the cost of a good or service offered to customers to increase trade.
Offering discounts enhances sales but not profitability. So, there is a trade-off that must be considered properly.
Calculations:Ordinarily, if the coupon price is more than $100, the 10% off coupon becomes better than the $10 off coupon.
For instance, if the coupon price is $110, the 10% off coupon will yield a discount amount of $11 ($110 x 10%), which is more than $10.
For instance, if the coupon price is $99, the discount amount will be $9.90, which is less than $10.
Thus, a 10% off coupon and a $10 off coupon do not offer the same discount amount unless the list price is $100, no more, no less.
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Question Completion:Compare the discounts you would receive with a 10% off coupon versus a $10 off coupon.
Maricopa's Success scholarship fund receives a gift of $ 215000. The money is invested in stocks, bonds, and CDs. CDs pay 2.25 % interest, bonds pay 2 % interest, and stocks pay 9.2 % interest. Maricopa Success invests $ 15000 more in bonds than in CDs. If the annual income from the investments is $ 8087.5 , how much was invested in each account?
The amount of money invested in stocks was $50000, $90000 was invested in bonds and $75000 was invested in CDs
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
An independent variable is a variable that does not depend on other variables while a dependent variable is a variable that depends on other variables.
Let x represent the amount of money invested in stocks, y represent bonds and z represent CDs, hence:
x + y + z = 215000 (1)
Also:
0.0225z + 0.02y + 0.092x = 8087.5 (2)
And:
y = 15000 + z (3)
The solution of the equation is:
x = 50000, y = 90000 and z = 75000
The amount of money invested in stocks was $50000, $90000 was invested in bonds and $75000 was invested in CDs
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A rectangular dining room has an area of 64 m². Its perimeter is 32 m. What are the dimensions of the dining room
Given the area and perimeter of the rectangular dining room, the dimensions of the dining room are 8m by 8m.
What are the dimensions of the dining room?The area of rectangle is expressed as;
A = l × b
The perimeter of a rectangle is expressed as;
P = 2( l + b )
Given that;
Area of the rectangular dining room A = 64m²Perimeter of the rectangular dining room P = 32mLength of the rectangular dining room l = ?Breadth of the rectangular dining room b = ?We use the information given to form a system of equations.
A = l × b
64 = l × b
P = 2( l + b )
32 = 2( l + b )
Hence, we have;
64 = l × b --------- equ 1
32 = 2( l + b )-----equ 2
Lets make l the subject of the formula in equation 1
64 = l × b
l = 64/b -------------equ 3
Substitute equation 3 into equation 2
32 = 2( l + b )
32 = 2l + 2b
32 = 2(64/b) + 2b
32 = 128/b + 2b
Now, we solve for b.
32 = 128/b + 2b
32 - 2b = 128/b
32b - 2b² = 128
2b² - 32b + 128 = 0
We factor out 2
2( b² - 16b + 64) = 0
From perfect square rule
2( 8-b )² = 0
divide both side by 2 and take the square root of both side
( 8-b )² = 0
8-b = 0
b = 8
Substitute b=8 into equation 1
64 = l × b
64 = l × 8
l = 64/8
l = 8
Given the area and perimeter of the rectangular dining room, the dimensions of the dining room are 8m by 8m.
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The graph of a function g is shown below.
Use the graph of the function to find its average rate of change from x=-2 to x=4.
Simplify your answer as much as possible.
1-10
Answer: -2
Step-by-step explanation:
The graph passes through (-2, 3) and (4, -9).
So, the average rate of change is [tex]\frac{-9-3}{4-(-2)}=\boxed{-2}[/tex]
A:HOW MANY DISTINCT ARRANGEMENTS CAN BE MADE WITH THE LETTERS IN THE
WORD CONNECTION?
B: HOW MANY AUTOMOBILE LICENSE PLATES CAN BE MADE IF EACH PLATE
CONTAINS 3 DIFFERENT DIGITS FOLLOWED BY 3 DIFFERENT LETTERS?
Answer:
A:The Reqd. No. of Permutations=3360
Step-by-step explanation:
I couldn’t find the answer to B..sorry
G wants to get over 9000 comments on his videos. He gets the same number of comments for each cat video that he uploads, and he gets the same number of comments for each dog video that he uploads. Let C represent the number of cat videos and D represent the number of dog videos that G should upload to achieve his goal.
750C+450D>9000 According to the inequality, how many comments does each cat video get, and how many comments does each dog video get?
can G achieve his goal by uploading 8 cat videos and 7 dog videos?
Answer:
Each Cat video gets 750 comments
Each dog video gets 450 comments
Hope this helps you!
Answer:
He gets 750 comments on the cat videos and 450 on the dog videos the answer to the second question is yes
Step-by-step explanation:
khan :)
To win at LOTTO in one state, one must correctly select 7 numbers from a collection of 61 numbers (1 through 61). The order in which the selection is made does not matter. How many different selections are possible?
Using the combination formula, it is found that 436,270,780 different selections are possible.
What is the combination formula?[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by:
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In this problem, 7 numbers are taken from a set of 61, hence the number of different selections is given by:
C(61,7) = 61!/(7! x 54!) = 436,270,780
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What is the resulting polynomial when -7w² +13 is subtracted from -2w²+6w - 19?
Answer:
5w^2 + 6w - 32
Step-by-step explanation:
Subtract as needed and combine like terms
(-2w^2 + 6w - 19) - (-7w^2 +13) dont forget to distribute the negative
so you get this:
-2w^2 + 7w^2 = 5w^2
6w
-19 - 13 = - 32
your answer is 5w^2 + 6w - 32
hope this helps :)
Hi :)
Let's solve by combining like terms
——————————Combining like terms means you add or subtract
terms with the same variable (if any)terms with the same exponentSolve by combining like terms
[tex]\implies\darkblue\sf{-2w^2+6w-19-(-7w^2+13)}\\\implies\darkblue\sf{-2w^2+6w-19+7w^2-13}\\\implies\darkblue\sf{-2w^2+7w^2+6w-19-13}\\\implies\darkblue\sf{\underline{5w^2+6w-32}}[/tex]
[tex]\tt{Learn\:More;Work\:Harder}[/tex]
:)
Suppose you are still committed to owning a $190,000 BMW (see Question 9). If you believe your mutual fund can achieve a 12% annual rate of return and you want to buy the car in nine years on the day you turn 30, how much must you invest today?
The amount of money which you must invest today is equal to $68,517.85.
How to calculate the present value?Mathematically, the present value of an investment can be calculated by using this formula:
Present Value, PV = FV/(1 + r)^t
Given the following data:
Time, t = 9 years.
Future value, FA = $190,000.
Interest rate, r = 12% = 0.12.
Substituting the given parameters into the formula, we have;
PV = 190,000/(1 + 0.12)^9
PV = 190,000/(1.12)^9
PV = 190,000/2.7730
PV = $68,517.85.
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Use the formula to find the standard error of the distribution of differences in sample means, x¯1-x¯2. Samples of size 100 from Population 1 with mean 90 and standard deviation 11 and samples of size 85 from Population 2 with mean 71 and standard deviation 15 Round your answer for the standard error to two decimal places. standard error = Enter your answer in accordance to the question statement
Using the Central Limit Theorem, the standard error of the distribution of differences in sample means is of 1.97.
What is the standard error for each sample?According to the Central Limit Theorem, it is given by the standard deviation of the sample divided by the square root of the sample size, hence:
s1 = 11/sqrt(100) = 1.1.s2 = 15/sqrt(85) = 1.63.What is the square root of the distribution of differences?It is given by the square root of the sum of the standard errors of each sample squared, hence:
s = sqrt(1.1² + 1.63²) = 1.97.
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Can anyone help me with this question!
Answer:
94. 25 in³Step-by-step explanation:
The radius of the base is the distance between the center of the circle and the edge of the base, therefore in this case it is equal to 3 in.
The volume of a cone is given by:
V = π * r² * h / 3
V = π * (3)² * 10 / 3
V = 94. 25 in³
Therefore, The volume of is 94. 25 in³.
Step-by-step explanation:
6.25 points
Find the GCF for the list.
10m4, 40m⁹
10m5
40m4
10m4
6.25 points
list
A taxicab charges $1.75 for the flat fee and $0.25 for each mile. Write a
O $1.75 + $0.25x ≤ $15
O$1.75 + $0.25x ≥ $15
$0.25 + $1.75x ≤ $15
$0.25 + $1.75x ≥ $15
Calculus HW , will someone please teach me how to do this problem. thanks! 10 Points
Your answers seem to be on the right track. These online homework apps can be picky about the answer they accept, though.
Given [tex]f(x) = x^4 - 2x^2 + 3[/tex], we have derivative
[tex]f'(x) = 4x^3 - 4x = 4x (x^2 - 1) = 4x (x - 1) (x + 1)[/tex]
with critical points when [tex]f'(x) = 0[/tex]; this happens when [tex]x=0[/tex] or [tex]x=\pm1[/tex].
We also have second derivative
[tex]f''(x) = 12x^2 - 4 = 12 \left(x^2-\dfrac13\right) = 12 \left(x - \dfrac1{\sqrt3}\right) \left(x + \dfrac1{\sqrt3}\right)[/tex]
with (possible) inflection points when [tex]x=\pm\frac1{\sqrt3}=\pm\frac{\sqrt3}3[/tex].
Intercept
If "intercept" specifically means [tex]y[/tex]-intercept, what you have is correct. Setting [tex]x=0[/tex] gives [tex]f(0) = 3[/tex], so the intercept is the point (0, 3).
They could also be expecting the [tex]x[/tex]-intercepts, in which case we set [tex]f(x)=0[/tex] and solve for [tex]x[/tex]. However, we have
[tex]x^4 - 2x^2 + 3 = \left(x^2 - 1\right)^2 + 2[/tex]
and
[tex]x^2-1\ge-1 \implies (x^2-1)^2 \ge (-1)^2 = 1 \implies x^4-2x^2+3 \ge 2[/tex]
so there are no [tex]x[/tex]-intercepts to worry about.
Relative minima/maxima
Check the sign of the second derivative at each critical point.
[tex]f''(-1) = 8 > 0 \implies \text{rel. min.}[/tex]
[tex]f''(0) = -4 < 0 \implies \text{rel. max.}[/tex]
[tex]f''(1) = 8 > 0 \implies \text{rel. min.}[/tex]
So we have two relative minima at the points (-1, 2) and (1, 2), and a relative maximum at (0, 3).
Inflection points
Simply evaluate [tex]f[/tex] at each of the candidate inflection points found earlier.
[tex]f\left(-\dfrac{\sqrt3}3\right) = \dfrac{22}9[/tex]
[tex]f\left(\dfrac{\sqrt3}3\right) = \dfrac{22}9[/tex]
Using the standard form of the equation in question 1, what are the length of the radius and the coordinates of the center for this particular circle? Watch your signs for the variables h and k. 12pt
The coordinates of the center is at (1, 0) and the length of the radius is 4 units.
What is an equation?An equation is an expression that shows the relationship between two or more variables and numbers.
The standard form of a circle is:
(x - h)² + (y - k)² = r²
Where (h, k) is the center of the circle and r is the radius.
The coordinates of the center is at (1, 0) and the length of the radius is 4 units.
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#SPJ1
Write y(t)=2sin 4 pi t + 5 cos 4 pi t) in the form y(t) A sin (wt + Ø) and identify the amplitude, angular frequency, and the phase shift of the spring motion.
Record your answers in the response box.
Expanding the desired form, we have
[tex]A \sin(\omega t + \phi) = A \bigg(\sin(\omega t) \cos(\phi) + \cos(\omega t) \sin(\phi)\bigg)[/tex]
and matching it up with the given expression, we see that
[tex]\begin{cases} A \sin(\omega t) \cos(\phi) = 2 \sin(4\pi t) \\ A \cos(\omega t) \sin(\phi) = 5 \cos(4\pi t) \end{cases}[/tex]
A natural choice for one of the symbols is [tex]\omega = 4\pi[/tex]. Then
[tex]\begin{cases} A \cos(\phi) = 2 \\ A \sin(\phi) = 5 \end{cases}[/tex]
Use the Pythagorean identity to eliminate [tex]\phi[/tex].
[tex](A\cos(\phi))^2 + (A\sin(\phi))^2 = A^2 \cos^2(\phi) + A^2 \sin^2(\phi) = A^2 (\cos^2(\phi) + \sin^2(\phi)) = A^2[/tex]
so that
[tex]A^2 = 2^2 + 5^2 = 29 \implies A = \pm\sqrt{29}[/tex]
Use the definition of tangent to eliminate [tex]A[/tex].
[tex]\tan(\phi) = \dfrac{\sin(\phi)}{\cos(\phi)} = \dfrac{A\sin(\phi)}{\cos(\phi)}[/tex]
so that
[tex]\tan(\phi) = \dfrac52 \implies \phi = \tan^{-1}\left(\dfrac52\right)[/tex]
We end up with
[tex]y(t) = 2 \sin(4\pi t) + 5 \cos(4\pi t) = \boxed{\pm\sqrt{29} \sin\left(4\pi t + \tan^{-1}\left(\dfrac52\right)\right)}[/tex]
where
• amplitude:
[tex]|A| = \boxed{\sqrt{29}}[/tex]
• angular frequency:
[tex]\boxed{4\pi}[/tex]
• phase shift:
[tex]4\pi t + \tan^{-1}\left(\dfrac 52\right) = 4\pi \left(t + \boxed{\frac1{4\pi} \tan^{-1}\left(\frac52\right)}\,\right)[/tex]
Solve (72 )3 ÷73 and answer properly no spammers allowed
