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O WEEK 4: QUADRATIC EQUATIONS AND FUNCTIONS
Finding intercepts of a nonlinear function give
The graph of a function is given below.
Give all y-intercepts and x-intercepts shown.

(a) y-intercept(s):
(b) x-intercept(s):

If there is more than one answer, separate them with commas.

Therefore, The dilated image's vertices are as follows:B'(2 , 3)   C'(3 , 3)     D'(1 , 3)

It is not necessary for the dilation's center to be inside the form. It might be the polygon's vertices. The preimage is still a square in this case, but the center of the dilatation is at the top-left vertex, which is shared by both of the dilated pictures (one smaller and one larger).

To get the vertices of the dilated image, multiply each vertex's coordinate by a factor of three.

B(6,9)⇒B'(1/3×6,1/3×9) = B'(2 , 3)

C(9,9)⇒C'(1/3×9 , 1/3×9) = C'(3 , 3)

D(3,9)⇒D'(1/3×3 ,1/3×9) = D'(1 , 3)

The dilated image's vertices are as follows:

B'(2 , 3)   C'(3 , 3)     D'(1 , 3)

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Answer:

Step-by-step explanation:

1

The one to one functions in the options includes

A one-to-one function, also known as an injective function, is a type of function in mathematics where every element in the domain maps to a unique element in the range.

So we can say that, each input value corresponds to exactly one output value, and no two distinct input values have the same output value.

Graphically, one-to-one functions pass the horizontal line test, meaning that no horizontal line intersects the graph of the function more than once.

Applying the horizontal line test shows that the one-to-one function are: Graph 1, 4 and 6

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Answer:

range = 22

Step-by-step explanation:

the range is the difference between the largest and smallest values in the data set.

largest value = 25 , smallest value = 3

range = 25 - 3 = 22

25% percent of the females in Math town have brown eyes. To find the number of females with brown eyes, we need to subtract the number of males with brown eyes from the total number of people with brown eyes

what is percent  ?

A percent is a way of expressing a number as a fraction of 100. The symbol for percent is %, which means "per hundred".

In the given question,

Let's assume that there are 100 people in Math town.

60% of them are males, so there are 60 males and 40 females.

Out of the 60 males, 30% have brown eyes, so there are 18 males with brown eyes.

28% of the entire population have brown eyes, so there are 28 people with brown eyes.

To find the number of females with brown eyes, we need to subtract the number of males with brown eyes from the total number of people with brown eyes:

28 - 18 = 10

So, there are 10 females with brown eyes.

Out of the 40 females in Math town, what percentage have brown eyes?

10 brown-eyed females out of 40 females is:

10/40 = 0.25 or 25%

Therefore, 25% of the females in Math town have brown eyes.

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The amplitude and period of each function are calculated and listed below

The amplitude and period of each function can be determined using their standard form:

y = A sin (Bx + C) or y = A cos (Bx + C)

where |A| is the amplitude, B is the frequency (or angular frequency), and 2π/B is the period.

Let's apply this formula to each function:

y = cos(2x)

Amplitude = 1

Period = 2π/2 = π

y = -sin(2x)

Amplitude = 1

Period = 2π/2 = π

y = -3sin(3x)

Amplitude = 3

Period = 2π/3

y = (1/2)cos(4x)

Amplitude = 1/2

Period = 2π/4 = π/2

y = 10sin(1/2x)

Amplitude = 10

Period = 2π/(1/2) = 4π

y = 5cos(1/4x)

Amplitude = 5

Period = 2π/(1/4) = 8π

y = (-1/3)cos(1/3x)

Amplitude = 1/3

Period = 2π/(1/3) = 6π

y = 4sin(-2x)

Amplitude = 4

Period = 2π/(-2) = -π

y = -2sin(2πx)

Amplitude = 2

Period = 2π/(1) = 2π

y = -3sin(πx)

Amplitude = 3

Period = 2π/π = 2

y = 1 + (1/2)cos(πx)

Amplitude = 1/2

Period = 2π/π = 2

y = -2 + cos(4πx)

Amplitude = 1

Period = 2π/(4π) = 1/2

The graphs of the functions are added as attachment

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Answer: Vertical angles are formed by the intersection of two lines and are always equal in measure. Therefore, to find the angle that is vertical to angle ∠DEB, we need to look for another angle that has the same measure as ∠DEB.

Unfortunately, we do not know the measure of angle ∠DEB, so we cannot determine which angle is vertical to it based on the given information.

Therefore, the correct answer is: None of the angles can be determined as vertical to angle ∠DEB without additional information.

Step-by-step explanation:

Answer:

The answer for Probability is 1/2

Step-by-step explanation:

7,6,8,1,54,23

Probability of spinning an odd number greater than 3=1,7,23

P=3/6

P=1/2

The value of x and y are -3 and 3 respectively as shown in the graph

An equation is an expression that shows how numbers and variables using mathematical operators.

Given the equation:

2x - 3y = -13      (1)

Also:

4x + 2y = -2    (2)

From equation 2:

4x + 2y = -2

2y = -4x - 2

y = -2x - 1

substitute y = -2x - 1 in equation 1:

2x - 3(-2x - 1) = -13

2x + 6x + 3 = -13

8x = -16

x = -2

Put x = -2 in eqn 2:

4(-2) + 2y = -2

-8 + 2y = -2

2y = 6

y = 3

The value of x and y are -3 and 3 respectively

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Answer:

the volume of the cylinder is 904.32 m³

Step-by-step explanation:

Recall the general equation for a line in slope-intercept form:

[tex]y=mx+b[/tex]

where:

[tex]\text{y = y-coordinate}[/tex]

[tex]\text{m}=\bold{slope}[/tex]

[tex]\text{x = x-coordinate}[/tex]

[tex]\text{b = y-intercept}[/tex]

In this equation, if you isolate for [tex]y[/tex], you end up with an equation that looks like this:

[tex]-4x+y=0[/tex]

[tex]-4x +4x+y=0+4x[/tex]

[tex]\implies y=4x+0[/tex]

Recall that in direct variation, the graph goes through the origin, (0,0). Since the y-intercept of this equation is 0, this means the graph intersects the origin. Thus, this graph represents direct variation.

If you were to graph the line, you would see the intersection at (0,0):

graph[tex]\{y=4x [-12.66, 12.65, -6.33, 6.33]\}[/tex]

The requried, quadratic equation that models the path of the golf ball, is 16.0850t² - 110.560t - 190 = 0 and the angle the ball takes off is 64.4⁰.

The motion of the golf ball can be modeled using two equations, one for the vertical motion and another for the horizontal motion.

The equation for the vertical motion of the ball is given by:

h = V₀t - ¹/₂gt²,

Using the given values, we find that,
V₀ = √(2 x 32.17 x 190) = 110.56 ft/s.

Substituting this value and the value of g into the equation of motion for the vertical direction, we get -190 = 110.56t - ¹/₂(32.17)t², which simplifies to 16.0850t² - 110.560t - 190 = 0.

The equation for the horizontal motion of the ball is given by [tex]x = V_xt[/tex], where x is the horizontal distance traveled, Vx is the initial horizontal velocity, and t is the time. Using the given value of x, we can find Vx as [tex]V_x = 440/t.[/tex]

We can then solve the quadratic equation 16.0850t² - 110.560t - 190 = 0 to find the time t of flight.
Using the formula method, we get t = 8.3 s. Substituting this value into the equation for the horizontal motion, we get [tex]V_x = 440/8.3 = 53.01 ft/s.[/tex]

Finally, we can find the angle of projection θ using the formula tan θ = [tex](V_y) / (V_x)[/tex],

tan θ = (110.56)/(53.01),

which simplifies to tan θ = 2.086. Taking the inverse tangent of both sides, we get θ = tan⁻¹(2.086) = 64.4⁰.

Therefore, the angle at which the ball takes off is 64.4⁰.

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Answer:

  (b)  $21,495.64

Step-by-step explanation:

You want to know the value of $15,000 after it has earned continuously compounded interest at 17.99% for 2 years.

The value of an account earning interest at annual rate r compounded continuously is ...

  A = P·e^(rt)

where P is the initial account value, and t is the number of years.

In this problem, we have P=15000, r=0.1799, and t=2, so the amount due will be ...

  A = $15,000·e^(0.1799·2) ≈ $21,495.64 . . . choice B

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Answer:

(a) The total ratio is 1 + 3 = 4, so Jill's share is (3/4) of £60:

Jill's share = (3/4) × £60 = £45.

(b) The ratio of their ages remains the same, so if Jack is one year older, Jill is also one year older. Therefore, Jill's age is 4 and Jack's age is 2, and Jack's share is (2/4) of £60:

Jack's share = (1/2) × £60 = £30.

(c) Let x be the age of Jack when he receives £25 from Uncle Fred. According to the problem, Jill is always two years older than Jack. Therefore, when Jack is x years old, Jill is x + 2 years old. The ratio of their ages is x : (x + 2), and this ratio is equal to 5/7 (25 pounds is 5/7 of 35 pounds, which is the total amount that they will receive). We can write an equation to solve for x:

x/(x+2) = 5/7

7x = 5(x + 2)

7x = 5x + 10

2x = 10

x = 5

Therefore, Jack will be 5 years old when he receives £25 from Uncle Fred at Christmas.

A) £45 because Jill will get 3/4 and Jack will get 1/4. To check our work, 60 divided by 4 is 15. 15 times 3 is 45 and 15 times one is 15.

B) £30 because it is a new year, both nephews will increase by one year. Now Jill is 4 and Jack is 2. So each kid will get half the share. 2 divided by 4 is 0.5 which is half of one. Half of 60 is 30.

C) 5 years Jill will always be 2 years older than Jack; this shows us that if some more years pass, Jill will be 7 and Jack will be 5 —> 1/4 x £60 = £25 Multiplying both sides by 4/60, we get:x = 5/4. This means Jack will be 5.

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Please remember to revise this and make it in your own words if you want! I am unsure about question C so comment!! I hope this helps you. -Doodle

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The value of  p-value is (a)   0.0025. (b). The decision regarding the null hypothesis is to  Reject the H0 (c.) Yes there is a difference in the mean given by the two offices.

Recall that The p-value or probability value is a number, calculated from a statistical test, that describes how likely your results would have occurred if the null hypothesis were true.

The first thing is to find the difference between the set of data

30 - 26 = 4

22 - 19 = 3

25 - 20 = 5

19 - 15 = 4

26 - 19 = 7

The find the average of the set of scores

24.4 , 19.8, 1.52

Hypothesis formulation

H0: µ1 = µ2

Ha: µ1 ≤ µ2

Decide the test statistic and the level of significance:

alpha = 0.05

df = 5 - 1 = 4

The upper critical value at the level of significance = 2.7764

And the lower critical value at the level of significance = -2.7764

The decision rule

Reject H0 if |t| > 2.7764

Test statistics calculating

bar x1 - bar x2 =

24.4 - 19.8 = 4.6

standard error = s / √n

= 0.6798

t statistic = 4.6 / 0.6798

= 6.767

The decision would be given that 6.7670 > 2.7764, we would have to reject the null hypothesis and accept the alternate hypothesis.

The p value is given as 0.0025. This is the probability of rejecting the true H0

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Matching each function with the transformation that took place gives:

g(x) = -(x - h)ⁿ: Reflection across the x-axis

j(x) = axⁿ + k: Translation of k units up

m(x) = (x + h)ⁿ - k: Translation of h units to the left

Vertical dilation: Functions of form f(x) = 2ⁿ are vertically dilated.

In mathematics, a transformation refers to a process of changing the size, position, or orientation of a geometric object or a function.

Transformations are usually expressed as a set of rules or equations that determine how each point or element in the original object or function maps to a corresponding point or element in the transformed object or function. Common types of transformations include translations, rotations, reflections, dilations, and combinations of these operations.

Transformations are widely used in many areas of mathematics, physics, engineering, and computer science to study and model complex systems and phenomena.

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In the ratio, the odds that are less generous is C) 19:1 and D)18:1.

What is ratio?

When two numbers are compared, the ratio between them shows how often the first number contains the second. As an illustration, the ratio of oranges to lemons in a dish of fruit is 8:6 if there are 8 oranges and 6 lemons present.

Here odds for bets are 11:1.

That means if for every $1  you bets, you may wins $11.

There are [tex]\frac{1}{1+11}=\frac{1}{12}=8.33\%[/tex] chance of this happens.

If the friends thinks odds are too generous , then the odds that less than generous are,

=> for 4:1 then [tex]\frac{1}{1+4}=\frac{1}{5}= 20\%[/tex]

=> For 6:1 then [tex]\frac{1}{6+1}=\frac{1}{7}= 14\%[/tex]

=> For 19:1 then [tex]\frac{1}{19+1}=\frac{1}{20}=5\%[/tex]

=> For 18:1 then [tex]\frac{1}{18+1}=\frac{1}{19}=5.2\%[/tex]

Hence the odds that are less generous is C) 19:1 and D)18:1.

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Answer:

1600

Step-by-step explanation:

The answer you tried is equivalent.

The length of the arc with a central angle of 285 degrees and radius 16 inches is 79.6 inches

From the question, we have the following parameters that can be used in our computation:

central angle = 285 degrees

Radius = 16 inches

Using the above as a guide, we have the following:

Length of arc = central angle/360 * 2 * 3.14 * Radius

Substitute the known values in the above equation, so, we have the following representation

Length of arc = 285/360 * 2 * 3.14 * 16

Evaluate

Length of arc = 79.6

Hence, the length of the arc is 79.6 inches

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Here is an example of how to create an object model, UML diagrams, and contract obligations in general.

Let's have a look at a basic "Bank Account" system, where a customer may deposit money, withdraw it, and see the amount of their account. The following classes would be represented in the UML class diagram for this system:

Bank Account Client

The following characteristics would be included in the Bank Account class:

string Account Number

stability: float

and the subsequent techniques:

deposit(float amount) = None

withdrawal(float :amount) = None

float: check Balance()

The following characteristics would apply to the Customer class:

string name

list of accounts[Bank Account]

and the subsequent techniques:

account: Bank Account; add account(account) -> None

account: Bank Account; remove  Account(account: None)

Bank Account: get  Account(Account number: string)

For the Bank Account class, the design by contract obligations would be:

Pre-conditions:

amount > 0

withdraw(amount): amount > 0 and balance >= amount

Check Balance(): None

Post-conditions:

deposit(amount): balance += amount

withdraw(amount): balance -= amount

Check Balance(): return balance

The design by contract responsibilities for the Customer class would be:

Pre-conditions:

Add account(account): account not in accounts

Post-conditions:

Add account(account): accounts. append(account)

remove (account): accounts.

get account (account number): return the Bank Account object with the given account Number.

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The calculated degrees of freedom for the t-test is 18

The degrees of freedom (df) for an independent t-test with two samples can be calculated using the following formula:

df = (n1 + n2) - 2

Where n1 is the sample size of the first group and n2 is the sample size of the second group.

In this case, the first sample has N = 8 and the second sample has N = 12.

Therefore, the degrees of freedom for the t-test will be:

df = (8 + 12) - 2

df = 18

So the t-test for this experiment will have 18 degrees of freedom.

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A circle-based model is highly suitable for the design and construction of roundabouts, which present themselves as practical solutions to tackle traffic-related issues.

Roundabout intersections involve vehicles circulating around a central island in a counterclockwise direction. The circular form guarantees fluidity in traffic movements and significantly reduces the requirement for traffic lights and stops signs while enhancing security.

This model entails using a circle, ascertaining that the diameter corresponds with the width of the roadway, with its center accommodating the location of the island at equidistant distances from all contact points on the circle's periphery, thus ensuring optimal functionality.

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Answer:

The odds that are less generous are 19:1, 14:1, and 15:1.

Step-by-step explanation:

You have to see which numbers are greater than 11.

Answer:

x = [tex]\frac{53}{40}[/tex]

Step-by-step explanation:

18 7/8 + __ = 20 1/5

18 7/8 = 151/8

20 1/5 = 101/5

[tex]\frac{151}{8}[/tex] + x = [tex]\frac{101}{5}[/tex]

x = [tex]\frac{101}{5}[/tex] - [tex]\frac{151}{8}[/tex]

x = [tex]\frac{53}{40}[/tex]

So, the answer is [tex]\frac{53}{40}[/tex]

A registered investment account, or RRSP, enables you to put money aside for retirement while delaying taxes on investment gains.

An annuity is a financial product that provides regular payments over a specified period of time in exchange for a lump sum of money. It's often used as a retirement income source and comes in different types such as fixed, variable, immediate, and deferred annuities. It's important to understand the terms, fees, and charges before purchasing an annuity.

Registered Retirement Savings Plan is the full form of RRSP.

In Canada, both workers and independent contractors can save and invest during their retirement through Registered Retirement Savings Plans (RRSP).

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2. The solution to the integral is: -2cot(x) - 4ln|sin(x)| - 3x + C

6. The integral converges and its value is 0.

2. To solve this integral using partial fractions, we need to rewrite the integrand as a sum of simpler fractions. We first rewrite the integrand as follows:

(2cos(x)/sin²(x) - 4 sin(x) - 5) = 2/sin(x)² - 4sin(x) - 5

Now we need to express the first term 2/sin(x)² as a sum of two fractions with different denominators. To do this, we let:

2/sin(x)² = A/sin(x) + B/sin(x)²

Multiplying both sides by sin(x)², we have:

2 = A sin(x) + B

Next, we need to determine the values of A and B. To do this, we can substitute two different values for x that make sin(x) ≠ 0, such as x = 0 and x = π/2.

Substituting x = 0, we get:

2 = A(0) + B

2 = B

Substituting x = π/2, we get:

2 = A(1) + B

2 = A + B

2 = A + 2

Therefore, A = 0.

So we have:

2/sin(x)² = B/sin(x)²

B = 2

Substituting these values back into our original expression, we get:

(2cos(x)/sin²(x) - 4 sin(x) - 5) = 2/sin(x)² - 4sin(x) - 5

= 2/sin(x)² - 4sin(x)/sin(x) + 2sin(x)/sin(x) - 5

= 2/sin(x)² - 4cot(x) + 2 - 5

= 2/sin(x)² - 4cot(x) - 3

Now we can integrate each term separately:

∫2/sin(x)² dx = -2cot(x) + C1

∫-4cot(x) dx = -4ln|sin(x)| + C2

∫-3 dx = -3x + C3

So the solution to the integral is:

∫(2cos(x)/sin²(x) - 4 sin(x) - 5) dx = -2cot(x) - 4ln|sin(x)| - 3x + C

where C = C1 + C2 + C3 is the constant of integration.

6. We can solve this integral by using the substitution u = x - 2, which gives us du = dx. We can then rewrite the integral as:

∞

∫₁ (1/(x-2)²) dx = ∞

∫-∞ (1/u²) du

Now, we can evaluate this integral as:

∞

∫-∞ (1/u²) du = [-1/u]∞

= [0 - (-1/(-∞))]

= 0 - 0

= 0

Therefore, the integral converges and its value is 0.

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The text format of the questions:

2. Use the method of partial fractions to find the integral:

∫(2cos(x)/sin²(x) - 4 sin(x) - 5) dx

6. State whether the integral converges or diverges. If it converges give the value it converges to.

∞

∫₁ (1/(x-2)²) dx

Answer:

To determine the classification of a triangle based on its side lengths, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the lengths of the legs (the shorter sides) is equal to the square of the length of the hypotenuse (the longest side).

For this particular triangle with side lengths 6 cm, 10 cm, and 12 cm, we can apply the Pythagorean theorem to determine whether the triangle is acute (all angles are less than 90 degrees), obtuse (one angle is greater than 90 degrees), or right (one angle is equal to 90 degrees).

Using the Pythagorean theorem, we find:

6^2 + 10^2 = 136

12^2 = 144

Since 136 is less than 144, we know that 6 cm, 10 cm, and 12 cm are the side lengths of a triangle that is obtuse (one angle is greater than 90 degrees), because the longest side (the hypotenuse) is greater than the sum of the squares of the lengths of the other two sides.

Therefore, the correct answer is: obtuse, because 6^2 + 10^2 < 12^2.