The binomial expansion is solved and the error in Dakota's statement is the incorrect substitution of 36g^2h^2 for the correct expression
Given data ,
Dakota made a mistake because the third term of the expansion of (2g + 3h) should have been 36g2h2. The binomial theorem asserts that the expansion of (2g + 3h) is as follows:
( x + y )ⁿ = ⁿCₐ ( x )ⁿ⁻ᵃ ( y )ᵃ
Here, x = 2g and y = 3h. Since term numbers begin at 0, since we are seeking for the third term, r = 2.
So , on simplifying the equation , we get
= nC2 * (2g)⁽ⁿ⁻²⁾ * (3h)²
= (n! / (2! * (n - 2)!)) * (2g)⁽ⁿ⁻²⁾ * (3h)²
= ((n * (n - 1)) / 2) * (2g)⁽ⁿ⁻²⁾ * (3h)²
Hence , the correct expression for the third term of the expansion of (2g + 3h) is ((n * (n - 1)) / 2) * (2g)^(n - 2) * (3h)², where n is the exponent in the binomial expansion.
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Citywide Delivery Inc. budgets $190 per month for telephone costs, which include the 3% federal excise tax. Citywide uses approximately 4,000 minutes per month, so the company signed up for the $96.00 plan. What is the maximum number of lines Citywide can sign up for and stay within their budget? Use the figure below:
Since we can't have a fractional number of lines, Citywide can sign up for a maximum of 1 line and still stay within their budget.
Let's start by calculating the total amount Citywide pays for their telephone costs without the federal excise tax. We can do this by subtracting the 3% tax from the total budget:
$190 / 1.03 = $184.47
The $96 budget plan includes 4,000 minutes, which means that each minute costs:
$96 / 4,000 = $0.024
If Citywide wants to stay within their budget, they need to make sure that the total cost of the lines they sign up for plus the federal excise tax is less than or equal to $190.
Let's represent the number of lines Citywide signs up for as "n". The total cost of the lines plus tax can then be represented as:
n * $96 * 1.03
Setting this expression less than or equal to $190, we get:
n * $96 * 1.03 <= $190
Simplifying, we get:
n <= $190 / ($96 * 1.03)
n <= 1.945
Thus, Citywide can sign up for a maximum of 1 line and still stay within their budget.
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The volume and radius of a cylinder are given below.
V=157.08 ft.³
r = 5 ft.
What is the area of the base of the cylinder?Use the approximation π 3.1416 or the calculator in your calculations. Round your answer to the nearest hundredth.
What is the height of the cylinder?
Round your answer to the nearest hundredth.
The area of the base of the cylinder is 78.54 ft² and height of the cylinder is 6.31 ft.
To find the area of the base of the cylinder, we can use the formula:
A = πr²
Substituting the given value of radius, we get:
A = π(5 ft)²
A = 78.54 ft²
Therefore, the area of the base of the cylinder is 78.54 ft².
b. To find the height of the cylinder, we can use the formula for the volume of a cylinder:
V = πr²h
Substituting the given values of volume and radius, we get:
157.08 ft³ = π(5 ft)²h
Simplifying, we get:
h = 157.08 ft³ / (π(5 ft)²)
h = 6.31 ft
Therefore, the height of the cylinder is approximately 6.31 ft.
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Find the surface area of the figure. Round the the
nearest hundredth when necessary.
11 ft
3 ft
7.7 ft
10 ft
9 ft
The surface area of the figure is 174.7 sq. ft.
Consider the bottom rectangular surface of the figure.
The dimensions of the bottom rectangular surface are: length = 10 ft and width = 3 ft
Using the formula for the area of rectangle, the area of bottom surface of the figure would be,
A₁ = length × width
A₁ = 10 × 3
A₁ = 30 sq. ft.
The dimensions of the right rectangular surface are length = 9 ft and width = 3 ft
Using the formula for the area of rectangle, the area of bottom surface of the figure would be,
A₂ = length × width
A₂ = 9 × 3
A₂ = 27 sq. ft.
The dimensions of the left rectangular surface are length = 11 ft and width = 3 ft
Using the formula for the area of rectangle, the area of bottom surface of the figure would be,
A₃ = length × width
A₃ = 11 × 3
A₃ = 33 sq. ft.
There area two parallel triangular suface with dimensions: base = 11 ft and height = 7.7 ft
Using the formula of the area of triangle the surface area of these two triamgular surfaces would be,
A₄ = 2 × 1/2 × base × height
A₄ = base × height
A₄ = 11 × 7.7
A₄ = 84.7 sq. ft.
So, the total surface area of the figure would be,
A = A₁ + A₂ + A₃ + A₄
A = 30 + 27 + 33 + 84.7
A = 174.7 sq. ft.
This is the required surface area of the figure.
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Assume a triangle ABC has standard labeling. Determine whether SAA,ASA,SSA,SAS,SSS is given and then whenever the law of cosine or the law of sines should be used to solve the triangle
In the given triangle ABC,
i. SAS is given
ii. The law of cosine can be used to solve it.
What is the cosine rule?The cosine rule is a mathematical principle that can be applied to determine the unknown length of side, or measure of the internal angle of a none right angled triangle.
The cosine rule states that;
[tex]c^{2}[/tex] = [tex]a^{2}[/tex] + [tex]b^{2}[/tex] - 2abCos C
Considering the given attachment to the question, for a standard labelling of triangle ABC. It can be observed that;
a. The given properties of the triangle are: Side-Angle-Side (SAS)
b. The angle given is an included angle, so that the cosine law is required to solve the triangle.
Therefore, the answer is: SAS, law of cosine
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When a tank is 1/2 full it contains 45 liters of water. The area of the
base is 450 cm². What is the height of the tank?
When a tank is 1/2 full it contains 45 liters of water, the area of the base is 450 cm², the height of the tank is 200 cm.
Let the height of the tank be 'h' cm. Since the tank is half full, it contains 45 liters of water which is equal to 45,000 cubic cm.
The volume of water in the tank is given by:
Volume of water = (1/2) x (450 cm²) x (h cm)
Since the volume of water is 45,000 cubic cm, we can set up the following equation:
(1/2) x (450 cm²) x (h cm) = 45,000 cubic cm
Simplifying, we get:
225h = 45,000
h = 200 cm
In this case, the area of the base is given in square centimeters, so we must use cubic centimeters for the volume of water. Also, we converted the liters to cubic centimeters by multiplying by 1000 since 1 liter is equal to 1000 cubic centimeters.
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2. There is another surface that Molly does not need to paint, because it won’t show when she displays the model house. Describe that surface. (2 points)
Without additional information about the model house, it is impossible to accurately describe the surface that Molly does not need to paint. It could be any surface that will not be visible when the model house is displayed, such as the underside of a roof, the back of a wall, or the bottom of a floor.
In a choir, the number of girls is twice the number of bovs. The avera height of the boys is 150 cm and the average height of the girls is 159 cm
What is the average height of all the choir members?
Answer:
Step-by-step explanation: mark me brainliest
Find the area of triangle ABC with the given parts. Round to the nearest tenth as necessary
The area of the given triangle is expressed as: 25.6 in²
What is the area of the triangle?We are only given two sides of the triangle and an angle and so we must find the length of the third side to be able to find the area. The length of the third side is gotten from cosine rule to get:
a = √(14.1² + 7.2² - 2(14.1 * 7.2)*cos 30.3)
a = 8.68 in
In order for us to calculate the area of triangle which has 3 sides, we will have to utilize the Heron's Formula.
From the formula, the area of a triangle (A) that has 3 sides a, b, and c is calculated via the formula:
A = √[s(s - a)(s - b)(s - c)]
where:
s denotes the semi-perimeter of the specific triangle given by the formula: s = (a + b + c)/2.
s = (8.68 + 14.1 + 7.2)/2
s = 14.99 in
Thus:
Area = √[14.99(14.99 - 8.68)(14.99 - 14.1)(14.99 - 7.2)]
Area = 25.6 in²
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A teacher gives 6 students some cards to play a game
If teacher has total "52 cards", and teacher gives each student "1 card" until all 52 cards are distributed, then number of students which got exactly 9 cards are (b) 4 students.
There are 6 students and 52 cards. The teacher gives each student one card until all 52 cards are given out in game. We want to know how many students receive exactly 9 cards.
To solve the problem, we divide the total number of cards (52) by the number of students (6) to find the average number of cards per student.
⇒ 52 cards / 6 students = 8.67 cards per student;
Since we can't give a student a fraction of a card, we need to round down to 8 cards,
If we give each student 8 cards, that will total of 8 cards × 6 students = 48 cards. which leaves 4 cards left over that we can't give out evenly.
This means that four-students will receive 9 cards each and two students will receive 8 cards each.
Therefore, the correct option is (b).
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The given question is incomplete, the complete question is
A teacher gives 6 students some cards to play a game, She has 52 cards in total, The teacher gives each 1 card until all 52 cards are given.
How many students gets exactly 9 cards?
(a) 2
(b) 4
(c) 5
(d) 6.
the graph of y=g(x) is shown. draw the graph of y=g(-x)
To draw the graph of y = g(-x), we need to replace x with -x in the original function.
If y = g(x) is given by the following graph:
(see attachment labeled "Attachment1")
Then y = g(-x) can be obtained by reflecting the graph of y = g(x) about the y-axis:
(see attachment labeled "Attachment 2")
Therefore, the graph of y = g(-x) is the reflection of the graph of y = g(x) about the y-axis.
I hope this helps you! Mark brainliest if you can :D
The table shows the distribution, by age and gender, of the million people who live alone in a certain region. Use the data in the table to find the probability that a randomly selected person living alone in the region is in the 2534 age range.
The probability which is selected randomly from a lot of people living alone in the area in the 25-34 age range will be; 0.1487
Since the total digit of individuals living independently in the area = 31.6
The digit of individuals living in the area who fall within the 25 - 34 age range is 4.7
The probability formula will be
P = required outcome / Total possible outcomes
From the information provided above the given data is:
P(range 25 - 34) = 4.7 / 31.6
= 0.1487
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The answer choices are distributive property, associative property, communicative property
Answer:
Step 1: Distributive Property
Step 2: Distributive Property
Step 3: Commutative Property
Step-by-step explanation:
Step 1: This is distributing the (x + 4y + z) into the (x - 7y), so each term in the prior expression is multiplied by (x - 7y). Hence, Distributive Property.
Step 2: This is doing the same thing but to each individual term. The x in the first term is multiplied by each term in the (x - 7y), the 4y in the second term is multiplied by each term in the (x - 7y), and the same for z. Hence, Distributive Property.
Step 3: This is reordering the variables in the terms. For example, 4yx is changed to 4xy, and that is possible because of the Commutative Property. zx is also reordered into a xz and 7zy into a 7yz.
Feel free to ask any more questions. Hope this helps!!!
Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function.
y''-5y'=δ(t-1), y(0)=7, y'(0)=0.
a.) Find the Laplace transform of the solution.
b.) obtain the solution y(t)
c.)express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t=1.
The laplace transform is Y(s) = (7 + e^{-s}/s) / (s² - 5s)
The solution y(t) is y(t) = (-e^{5-t} + 2e^{-5} - 6(t-1) + 7u(t-1))/25.
The solution can be expressed as a piecewise-defined function:
y(t) = -e^{5-t}/25 + 2e^{-5}/25 - 6(t-1)/25 + 7/25, t < 1
y(t) = -e^{5-t}/25 + 2e^{-5}/25 - 6(t-1)/25 + 7/25 + (t-1)/25, t ≥ 1
We have,
a.)
To find the Laplace transform of the solution, we first take the Laplace transform of both sides of the differential equation using the initial value theorem, which states that the Laplace transform of the derivative of a function y(t) is sY(s) - y(0):
s²Y(s) - s y(0) - y'(0) - 5(sY(s) - y(0)) = e^{-s}
Simplifying and solving for Y(s), we get:
Y(s) = (7 + e^{-s}/s) / (s² - 5s)
b.)
To obtain the solution y(t), we use partial fraction decomposition to separate Y(s) into two terms:
Y(s) = A/(s-5) + B/s + C/(s-5)^2
Multiplying both sides by the denominator, we get:
7 + e^{-s}/s = A(s-5)² + Bs (s - 5) + C(s)
Setting s = 0, we get:
7 = 25A - 5B + 0C
Setting s = 5, we get:
7 + e^{-5}/5 = 0A + 5B + 5C
Taking the derivative with respect to s and setting s=0, we get:
0 = 10A - B
Solving these equations, we get:
A = -1/25
B = -2/5
C = 6/25
Therefore, the solution y(t) is:
y(t) = (-e^{5-t} + 2e^{-5} - 6(t-1) + 7u(t-1))/25
Where u(t) is the unit step function.
c.)
The solution can be expressed as a piecewise-defined function as follows:
y(t) = -e^{5-t}/25 + 2e^{-5}/25 - 6(t-1)/25 + 7/25, t < 1
y(t) = -e^{5-t}/25 + 2e^{-5}/25 - 6(t-1)/25 + 7/25 + (t-1)/25, t ≥ 1
At t = 1, there is a discontinuity in the first derivative of the solution due to the presence of the delta function in the original differential equation.
This causes a sudden jump in the slope of the graph of the solution.
Thus,
The laplace transform is Y(s) = (7 + e^{-s}/s) / (s² - 5s)
The solution y(t) is y(t) = (-e^{5-t} + 2e^{-5} - 6(t-1) + 7u(t-1))/25.
The solution can be expressed as a piecewise-defined function as:
y(t) = -e^{5-t}/25 + 2e^{-5}/25 - 6(t-1)/25 + 7/25, t < 1
y(t) = -e^{5-t}/25 + 2e^{-5}/25 - 6(t-1)/25 + 7/25 + (t-1)/25, t ≥ 1
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Assignment I 1. Prepare a questionare on Expenditures and Consumptions behaviours and attitudes of polytechnic Students (Case Study of TOPS)
Some questions that can be asked on the Expenditures and Consumptions behaviours and attitudes of polytechnic Students in a questionnaire are:
What is the average of your total proceeds each month?How much money, on average, do you disburse to purchase perishables and staples each month?Transit consumption - On an ordinary basis, how much are you spending periodically?To participate in reposeful activities or entertainments, monthly expenditure- How much would that amount to?When it comes to periodic expenses, do you have a pre-defined plan in paper?Do you rule over your costings every thirty days with rigorousness?Have any credits or financial obligations been dispensed for the satisfaction of debts?Utilizing a student loan, have you ever funded your current education or living outlays?How to make a questionnaire ?One must carefully consider the purpose of a questionnaire, as well as the information required before starting to design it. Also crucial are factors such as the target audience, number and type of questions (i.e., multiple-choice or open-ended), and its format (e.g., Likert scale).
The way in which questions are organized plays an important role, such that people can easily comprehend them and respond accurately while finishing the survey as fast as needed. Concludingly, logical sequencing of questions should be ensured for the overall design to make sense from beginning to end.
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For each equation complete the table of values and draw its graph for values of x from -1 to 3
Answer:
Step-by-step explanation:
Answer:
where is the question.
send it so I'll answer it
Use the relationship given in the right triangle and the inverse sine, cosine, and tangent functions to write 0 as a function of x in three different ways. It is not
necessary to rationalize the denominator.
a
-√√81-x²³.c-9
The measure of inverse of sinθ is c/√ (c² - a²)..
What is the measure of inverse of sinθ?The measure of inverse of sinθ is calculated by applying trigonometry identities for right triangles.
Mathematically, the trig identities are represented using the following method;
SOH CAN TOA
SOH is for sine θ
CAH is fof cos θ
TOA is for tan θ
From the diagram we need find the value of the opposite side;
b = √ (c² - a²)
Sin θ = b/c
The inverse of b/c = c/b
1/sinθ = c/b = c/√ (c² - a²).
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Help much needed pls and thank you.
Answer:
Step-by-step explanation:
A full revolution on a circle/radians is 2[tex]\pi[/tex], so keep adding or subtracting 2[tex]\pi[/tex] til you get a base angle, that's when the sign changes. Then decide which quadrant your in.
For this one the equivalent of 2[tex]\pi[/tex] is [tex]6\pi /3[/tex]
-29[tex]\pi[/tex]/3 + [tex]6\pi /3[/tex]
= -23[tex]\pi[/tex]/3 + [tex]6\pi /3[/tex]
= -17[tex]\pi[/tex]/3 + [tex]6\pi /3[/tex]
= -11[tex]\pi[/tex]/3 + [tex]6\pi /3[/tex]
= -5[tex]\pi[/tex]/3 + [tex]6\pi /3[/tex]
= 1[tex]\pi[/tex]/3 sign changed it's equavalent to [tex]\pi /3[/tex] Which is in the first quadrant. See unit circle.
-5[tex]\pi[/tex]/6 + [tex]12\pi /6[/tex]
=[tex]7\pi /6[/tex] third quadrant, i count quadrants by know [tex]\pi[/tex]/6 is 30°, so every 30° line is 1/6 of the unit circle. when i count 7 of them that's in the 3rd quadrant. Don't forget to count the axis's
2[tex]\pi[/tex]/3 is in the 2nd quadrant. Count my pi/3's which is 60°
45[tex]\pi[/tex]/7 - 14[tex]\pi[/tex]/7
= 31[tex]\pi[/tex]/7 - 14[tex]\pi[/tex]/7
=17[tex]\pi[/tex]/7 - 14[tex]\pi[/tex]/7
=3[tex]\pi[/tex]/7 - 14[tex]\pi[/tex]/7
= -11[tex]\pi[/tex]/7 1/7th's is not your typical unit circle angle
This one i think of logically. this is -1 4pi/7
1 pi going backwards is 180
keep going backards 4/7 is bigger than 1/2 so it's in the 1st quadrant
Please help show work 11 points
The solution to the system of equations in this problem is given as follows:
(0.6, 2.4).
How to solve the system of equations?The system of equations in the context of this problem is defined as follows:
y = 4x.y = -x + 3.Equaling the two equations, we can obtain the value of x, as follows:
4x = -x + 3
5x = 3
x = 3/5
x = 0.6.
Substituting the value of x into the second equation, we obtain the value of y as follows:
y = -0.6 + 3
y = 2.4.
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Exponential model. Further research of the new disease has confirmed that its spread is
not linear, but exponential. Experts have estimated it to double every 7 days when it is
first introduced to a population.
Suppose there were 3 cases reported on the morning of day 1 of the outbreak. How many
cases will there be by the end of day 21?
Answer:
Since the disease doubles every 7 days, the number of cases on day 21 will be 2^3 times the number of cases on day 14, which will be 2^2 times the number of cases on day 7, which will be 2^1 times the number of cases on day 1.
Starting with 3 cases on day 1, we can use the formula:
N = 3 * 2^(t/7)
where N is the number of cases after t days.
Plugging in t = 21, we get:
N = 3 * 2^(21/7) = 3 * 2^3 = 24
Therefore, there will be 24 cases by the end of day 21.
Step-by-step explanation:
Two machines, X and Y, produce earbuds. Let X represent the diameter of an earbud produced by machine X, and let
Y represent the diameter of an earbud produced by machine Y. X has a mean of 14 mm with a standard deviation of
0.6 mm, and Y has a mean of 15.2 mm with a standard deviation of 0.2 mm. Which answer choice correctly calculates
and interprets the mean of the difference, D = X-Y?
OD=-1.2; earbud manufacturers can expect the difference in the diameter of earbuds produced from machines X
and Y, on average, to be -1.2 mm.
OH = 0.4; earbud manufacturers can expect the difference in the diameter of earbuds produced from machines X and
Y, on average, to be 0.4 mm.
O = 1.2; earbud manufacturers can expect the difference in the diameter of earbuds produced from machines X and
Y, on average, to be 1.2 mm.
OD = 29.2; earbud manufacturers can expect the difference in the diameter of earbuds produced from machines X
and Y, on average, to be 29.2 mm.
-1.2, earbud manufacturers can expect the difference in the diameter of earbuds produced from machines X
Given that two machines, X and Y, produce earbuds
Let X represent the diameter of an earbud produced by machine X, and
let Y represent the diameter of an earbud produced by machine Y. X has a mean of 14 mm with a standard deviation of 0.6 mm
Y has a mean of 15.2 mm with a standard deviation of 0.2 mm.
The mean of the difference, D = X - Y, can be calculated as:
mean(D) = mean(X) - mean(Y)
= 14 - 15.2
= -1.2
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Find an equivalent equation in rectangular coordinates
Answer:
[tex]x^2+y^2 = 2x - 2y}[/tex]
Third answer option
Step-by-step explanation:
We are given the polar equation as
[tex]r = 2(\sin \theta - \cos \theta)[/tex]
and asked to convert it into rectangular form
We have the following equations which relate (r, θ) in polar form to (x, y) in rectangular form
[tex]\cos \theta=\dfrac{x}{r} \rightarrow x=r \cos\theta\\\\\sin \theta=\dfrac{y}{r} \rightarrow y=r \sin \theta\\\\[/tex]
[tex]r^2=x^2+y^2[/tex]
Original polar equation:
[tex]r = 2(\sin \theta - \cos \theta)[/tex]
Expand the right side:
[tex]r = 2\sin \theta - 2\cos \theta[/tex]
Substitute for sinθ and cosθ in terms of x and y
[tex]r & = 2\left(\dfrac{y}{r} - \dfrac{x}{r}\right)\\[/tex]
Multiply both sides by r:
[tex]r^2 = 2(x - y)\\r^2 = 2x - 2y\\[/tex]
Substitute [tex]r^2=x^2+y^2[/tex] on the left side:
[tex]\boxed{x^2+y^2 = 2x - 2y}[/tex]
This would be the third answer opton
What is the solution to this system?
(1, 0)
(1, 6)
(8, 26)
(8, –22)
Universal Exports Inc. is a small company and is considering a project that will require $700,000 in assets. The project will be financed with 100% equity. The company faces a tax rate of 25%. What will be the ROE (return on equity) for this project if it produces an EBIT (earnings before interest and taxes) of $140,000?
a. 16.50%
b. 15.00%
c. 11.25%
d. 12.00%
The ROE (return on equity) for this project if it produces an EBIT (earnings before interest and taxes) of $140,000 is given by A = 15 %
Given data ,
The Return on Equity (ROE) is calculated as the ratio of Net Income to Equity. Net Income is the EBIT (earnings before interest and taxes) minus the taxes, and Equity is the total assets minus the debt.
EBIT = $140,000
Tax rate = 25%
Total assets = $700,000
Debt = 0% (since the project is financed with 100% equity)
Taxes = Tax rate x EBIT
Taxes = 0.25 x $140,000
Taxes = $35,000
And , the net income is
Net Income = EBIT - Taxes
Net Income = $140,000 - $35,000
Net Income = $105,000
Now , the ROE is given by
ROE = (Net Income / Equity) x 100%
ROE = (Net Income / Total assets) x 100%
ROE = ($105,000 / $700,000) x 100%
ROE = 15%
Hence , the ROE is 15 %
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Math Algebra Help needed
You can use functions to complete the table as follow:
x f(g(x))
4 2
10 4
20 6
34 8
52 10
How to use functions to complete the table?A function is an expression that shows the relationship between the independent variable and the dependent variable. A function is usually denoted by letters such as f, g, etc.
We have:
f(x) = √(x+1)
g(x) = 2x - 5
When x = 4:
f(g(x)) = f(g(4))
f(g(4)) = f(2*4 - 5)
f(g(4)) = f(3)
f(g(4)) = √(3+1) [Remember f(x) = √(x+1)]
f(g(4)) = 2
When x = 10:
f(g(10)) = f(2*10 - 5)
f(g(10)) = f(15)
f(g(10)) = √(15+1)
f(g(10)) = 4
When x = 20:
f(g(20)) = f(2*20 - 5)
f(g(20)) = f(35)
f(g(20)) = √(35+1)
f(g(20)) = 6
When x = 34:
f(g(34)) = f(2*34 - 5)
f(g(34)) = f(63)
f(g(34)) = √(63+1)
f(g(34)) = 8
When x = 52:
f(g(34)) = f(2*52 - 5)
f(g(34)) = f(99)
f(g(34)) = √(99+1)
f(g(34)) = 10
Thus, fill the values into the table to complete it.
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Few families can survive on a single salary.
Please select the best answer from the choices provided
T
F
Few families can survive on a single salary is true statement
Many families rely on dual incomes to make ends meet, especially in areas with high living costs.
However, there are still many families that rely on a single salary to support themselves.
While it may be challenging to make ends meet on a single salary, it is still possible for families to survive and even thrive in these circumstances.
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Two forces F₁ and F₂ act on a particle P. The force F₁ is given by F₁ = (-i + 2j) N and F₂ acts in the direction of the vector (i + j). The resultant of F₁ and F₂ acts in the direction of the vector (i + 3j). The acceleration of Pis (3i + 9j) ms 2. At time t = 0, the velocity of Pis (31 -22j) m s´¯¹. Find the speed of P when t = 3 seconds. (4 marks)
The evaluated speed of P is 54.5 m s¯¹, under the condition that two forces F₁ and F₂ act on a particle P. The force F₁ is given by F₁ = (-i + 2j) N and F₂ acts in the direction of the vector (i + j).
To evaluate the speed of particle P when t = 3 seconds, we can apply the equations of motion formulas to calculate the velocity if the acceleration and initial velocity values are already given.
The formula can be derived :
v = u + at
Here,
v = final velocity of the particle,
u = initial velocity of the particle,
a = acceleration acting on the particle
t = time taken.
It is known to us that the acceleration of P is (3i + 9j) ms² and at time t = 0, the velocity of P is (31 -22j) m s¯¹. We can evaluate the initial velocity u by taking the magnitude of (31 -22j) that is
√(31² + (-22)²)
= 39.051 m s¯¹.
Now we can staging all values into the formula
v = u + at
v = 39.051 + (3i + 9j) × 3
v = 39.051 + (9i + 27j)
v = (39.051 + 9)i + (27)j
v = 48.051i + 27j
Then, when t = 3 seconds, the speed of particle P is √(48.051² + 27²)
= 54.5 m s¯¹.
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Paola has enough mulch to cover 48 square feet. She wants to use it to make three square vegetable gardens of equal sizes. Solve the equation 3s2 = 48 to find s, the length of each garden side (in feet).
The length of each garden side is 4 ft.
Given that, Paola has 48 ft² of mulch, she wants to make three square vegetable gardens of equal sizes, we need to find the length of each garden side.
Let s be the length of the side of the gardens,
Since, she need 3 gardens,
So,
3 × side² = 48
3s² = 48
s² = 16
s = 4
Hence, the length of each garden side is 4 ft.
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You want to purchase a new car in 4 years and expect the car to cost $76,000. Your bank offers a plan with a guaranteed APR of 4.5 %if you make regular monthly deposits. How much should you deposit each month to end up with $76,000 in 4years?
Monthly you should deposit approximately $43 to accumulate $76,000 in 4 years.
Given that,
f = $76,000
r = 4% = 0.04/12 = 0.0033
n = 4×12 = 48
We know that, a = (f×r)/((1+r)ⁿ⁻¹)
a is the annuity.
f is the future amount.
r is the interest rate per time period.
n is the number of time periods.
Here,
a = (76,000 ×0.0033)/((1.0033)⁴⁸⁻¹)
a = 76,000 ×0.0033×0.17132
a = 42.96
a ≈ $43
Therefore, monthly you should deposit approximately $43 to accumulate $76,000 in 4 years.
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Math Algebra Help needed
The composite functions are (f · g) (x) = (x + 2)² and (f / g) (x) = 1, whose domains are all real numbers. The value of each function at x = 2 is listed below:
(f · g) (2) = 16 (f / g) (2) = 1How to find and to analyze composite functions
In this problem we must determine and analyze composite functions. Composite functions can be found by combining two simpler functions thanks to operators. There are four operators to make composite functions:
Addition: (f + g) (x) = f(x) + g(x)Subtraction: (f - g) (x) = f(x) - g(x)Multiplication: (f · g) (x) = f(x) · g(x)Division: (f / g) (x) = f(x) / g(x)The domain of a function is the set of all possible of x-values.
First, we determine the two composite functions: f(x) = g(x) = x + 2.
Case A:
(f · g) (x) = (x + 2)²
Case B:
(f / g) (x) = 1
Second, determine the domain for each case:
Case A:
All real numbers.
Case B:
All real numbers.
Third, evaluate each case at x = 2:
Case A:
(f · g) (2) = (2 + 2)² = 4² = 16
Case B:
(f / g) (2) = 1
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a. The following is the input/output table for two industries X and Y. The values are in million of rupees.
Producers
X
Y
X
14
7
Users
Y
6
Final Demand
8
Total Output
28
18
11
36
Determine the outputs if the final demand changes to 20 for X and 30 for Y.
[3]
The outputs for industries X and Y will be 31 million and 43.2 million, respectively supposedly the final demand changes to 20 for X and 30 for Y.
How do we calculate?We will start with industry X.
The final demand for X hade an increment from 14 million to 20 million with an increase of 6 million, hence the final output of X must also increase by 6 million to make up for demand.
Users Final Producers
Demand X Y
X 14 7
Y 6 18
Total 20 25
The table tells us that for every additional million of final demand for X, the producers in industry X need to produce 0.5 million of output.
So the new total output for industry X is 28 + 3 = 31 million.
For industry Y
The same scenario occurred for Industry Y which had final output of 12 in order to meet the demand increment.
So the new total output for industry Y is 36 + 7.2 = 43.2 million.
In conclusion, the outputs for industries X and Y will be 31 million and 43.2 million, respectively, if the final demand changes to 20 for X and 30 for Y.
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