Prove tanAsinA + cosA = sec A

To verify if each given function is a solution of the differential equation, we need to substitute the function into the differential equation and check if the equation holds true.

For the differential equation y" - 4y = 0:

a) Substitute y(t) = e^(2t):

y" = (e^(2t))'' = 4e^(2t)

4y = 4e^(2t)

The equation y" - 4y = 0 holds true.

b) Substitute y(t) = cosh(t):

y" = (cosh(t))'' = cosh(t)

4y = 4cosh(t)

The equation y" - 4y = 0 holds true.

For the differential equation y" + 2y - 3y = 0:

a) Substitute y(t) = e^(-3t):

y" = (-3e^(-3t))

2y = 2e^(-3t)

The equation y" + 2y - 3y = 0 holds true.

b) Substitute y(t) = e^(t):

y" = e^(t)

2y = 2e^(t)

The equation y" + 2y - 3y = 0 holds true.

For the differential equation 12y" + 5ty' + 4y = 0, t > 0:

a) Substitute y(t) = t^2:

y" = 2

y' = 0

5ty' = 0

12y" + 5ty' + 4y = 12(2) + 0 + 4(t^2) = 24 + 4t^2

The equation 12y" + 5ty' + 4y = 0 holds true.

b) Substitute y(t) = t^2:

y" = 2

y' = 0

5ty' = 0

12y" + 5ty' + 4y = 12(2) + 0 + 4(t^2) = 24 + 4t^2

The equation 12y" + 5ty' + 4y = 0 holds true.

For the differential equation y" + y = sec(t):

a) Substitute y(t) = sec(t):

y" = sec(t)tan(t)

y = sec(t)

The equation y" + y = sec(t) holds true.

b) Substitute y(t) = sec(t):

y" = sec(t)tan(t)

y = sec(t)

The equation y" + y = sec(t) holds true.

Therefore, each given function is a solution to its respective differential equation.

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The percentage of category shelf facings for Brand F is 10% out of the total facings allotted for all brands in that category, based on the information provided.

To calculate the percentage of category shelf facings for Brand F, we need to determine the proportion of shelf facings allotted to Brand F out of the total facings allotted for all brands in that category.
Company U has 100 outlets, and half of those outlets carry Brand F. This means that there are 50 outlets that carry Brand F.
Out of the 50 facings allotted for all brands in that category, Company U allots Brand F 5 shelf facings.
To find the percentage, we divide the facings allotted to Brand F (5) by the total facings allotted for all brands in the category (50), and then multiply by 100 to express it as a percentage.
(5 facings / 50 facings) * 100 = 10%
Therefore, the percentage of category shelf facings for Brand F is 10%. This indicates that Brand F occupies 10% of the available shelf space in the category across Company U's outlets.



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Answer:

  an = 1/n^2

Step-by-step explanation:

We notice that the denominators are the squares of successive integers. The general term will be ...

  \(\boxed{a_n=\dfrac{1}{n^2}}\)

The difference quotient evaluated in x = 2 gives:

[ f(2 + h) - f(2)]/h = h + 9

So the last option is the correct one.

Here we know the function:

f(x) = x^2 + 5x + 6

And we want to find the difference quotient:

[ f(2 + h) - f(2)]/h

Let's evaluate the function:

f(2 + h) = (2 + h)^2 + 5*(2 + h) + 6

            = 2^2 + h^2 + 4h + 10 + 5h + 6

            = h^2 + 9h + 20

f(2) = 2^2 + 5*2 + 6

f(2) = 4 + 10 +  6 = 20

Then we have:

[ f(2 + h) - f(2)]/h

[h^2 + 9h + 20 - 20]/h

[h^2 + 9h ]/h

Taking the quotient we get:

[h^2 + 9h ]/h = h + 9

So the correct option is the last one.

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Answer:

Original price = $60

Step-by-step explanation:

Given:

Sales price of shoes = $45

Discount rate = 25%

Find:

Original price

Computation:

Original price = Sales price of shoes[1/(1-Discount rate)]

Original price = 45[1/(1-25%)]

Original price = 45[1/(1-0.25)]

Original price = 45[1/(0.75)]

Original price = $60

Answer:

Step-by-step explanation:

$60

it would be better to go to the online store

Answer:

66 dollars

Step-by-step explanation:

15+6=22

15×2=30

15+22+30=66

Answer:

Tim had 15 dollars. Paul had 6 dollars more than Tim. Rick had double Tim’s money. How much did they have ALTOGETHER is 66 dollars.

Step-by-step explanation:

→ Is know :

→ Asked :

How much did they have ALTOGETHER ?

→ Solution :

\( \bf{x =Tim's \: \: money + Paul's \: \: money + Rick's \: \: money }\)

\(\bf{x =15 + (15 + 6) + (2 \times 15) }\)

\(\bf{x =15 + 21 + 30 }\)

\(\bf{x =36 + 30 }\)

\(\bf{x =66 \: \: dollars }\)

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Answer:

6 + 6y + 6 = 12 + 6

Step-by-step explanation:

2(3) + 6y = 12

Multiply 2(3).

6 + 6y = 12

Subtract 6 on both sides, so y variable is isolated.

6 + 6y - 6 = 12 - 6

6y = 6

Divide by 6 on both sides.

6y/6 = 6/6

y = 1

System (1) is stable, causal, linear, time-invariant, and has memory. System (2) does not exhibit causality. However, it has memory due to the integration operation.

To determine the properties of the given systems, let's analyze each one separately:

System 1: y(t) = 4x(t - 5)cos(t)

1) Stability: A system is considered stable if its output remains bounded for bounded input. In this case, the output y(t) is multiplied by a cosine function, which is bounded between -1 and 1. As long as the input x(t) is bounded, the output will also be bounded. Therefore, System 1 is stable.

2) Causality: A system is causal if the output at any given time depends only on the current and past values of the input. In this case, the output y(t) depends on the current and past values of x(t) due to the term x(t - 5). Thus, System 1 is causal.

3) Linearity: A system is linear if it satisfies the properties of superposition and scaling. Superposition means that the output resulting from the sum of two inputs is equal to the sum of the outputs from each individual input. Scaling means that multiplying the input by a constant scales the output by the same constant. In this case, System 1 is linear since both superposition and scaling properties hold.

4) Time Invariance: A system is time-invariant if a time shift in the input results in the same time shift in the output. In this case, the term x(t - 5) indicates a time shift of 5 units. As a result, the system exhibits time invariance.

5) Memory: A system has memory if the output at any given time depends on past and/or future values of the input. In System 1, the output y(t) only depends on the current and past values of x(t), indicated by the term x(t - 5). Thus, it has memory.

System 2: y(t) = ∫x(t)dt - t

1) Stability: It depends on the properties and behavior of x(t) as well as the integration operation.

2) Causality: In this case, the output y(t) depends on both the current and future values of the input x(t) due to the integration operation. Therefore, System 2 is not causal.

3) Linearity: It depends on the properties and behavior of x(t) as well as the integration operation.

4) Time Invariance: It depends on the properties and behavior of x(t) as well as the integration operation.

5) Memory: The system has memory since the output y(t) depends on the integral of past and current values of x(t).

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Step-by-step explanation:

b) 3.14, 31.4, 0.314, 3 2/4, 3.141

3.14, 31.4, 0.314, 3.5, 3.141

0.314, 3.14, 3.141, 3.5, 31.4

Answer:

25 people

Step-by-step explanation:

c = 5p + 220

345 = 5p + 220

-220          -220

125 = 5p

/5       /5

25 = p

Answer:

5.76

Step-by-step explanation:

17.28/3

The information is not enough to prove that triangles DEF and OPQ are congruent through SAS, as we need two equal side lengths and one angle measure.

The Side-Angle-Side (SAS) congruence theorem states that if two sides of two similar triangles form a proportional relationship, and the angle measure between these two triangles is the same, then the two triangles are congruent.

The parameters given for this problem are given as follows:

As we are given only one side, instead of the two sides needed, we have that the information given is not enough to prove that the two triangles are congruent by the SAS congruence theorem.

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Maribella buys 434,285 pounds of sweet potatoes. To determine the number of pounds of sweet potatoes Maribella buys, we can set up a ratio based on the given information.

Given:

Weight of white potatoes = 55 pounds

Weight of sweet potatoes = 7887 times the weight of white potatoes

Let's denote the weight of sweet potatoes as SP.

Using the ratio, we can write the equation:

55 : SP = 1 : 7887

To solve for SP, we can cross-multiply:

55 * 7887 = 1 * SP

SP = 434,285 pounds

Therefore, Maribella buys 434,285 pounds of sweet potatoes.

Expressed as a mixed number:

Maribella buys 434,285 pounds of sweet potatoes, which can be written as 434,285 pounds.

Expressed as a fraction greater than one:

Maribella buys 434,285 pounds of sweet potatoes, which can be written as 434,285/1 pounds.

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Thus, beginning with a 1 = 5, the formula a n = a n-1 + 7 can be used to recursively find the nth term of the sequence.

A sequence in mathematics is an ordered collection of numbers that is typically defined by a formula or rule. Every number in the series is referred to as a term, and its location within the sequence is referred to as its index. Depending on whether the list of terms stops or continues indefinitely, sequences can either be finite or infinite. By their patterns or uniformity, sequences can be categorised, and the study of sequences is crucial to many areas of mathematics, such as calculus, number theory, and combinatorics. Mathematical, geometrical, and Fibonacci sequences are a few examples of popular sequence types.

given

The sequence's terms are all different by 7 (i.e., 12 - 5 = 19 - 12 = 26 - 19 =... = 7).

The following is a definition of a recursive formula for the nth element of the sequence:

a 1 = 5 (the first term of the series is 5) (the first term of the sequence is 5)

For n > 1, each term is derived by adding 7 to the preceding term, so a n = a n-1 + 7.

Thus, beginning with a 1 = 5, the formula a n = a n-1 + 7 can be used to recursively find the nth term of the sequence. For instance, we have

a_2 = a_1 + 7 = 5 + 7 = 12

a_3 = a_2 + 7 = 12 + 7 = 19

a_4 = a_3 + 7 = 19 + 7 = 26

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Answer:

\(17.2 {m}^{2} \)

Step-by-step explanation:

\((m + m)(8.6m) = (2m)(8.6m) = (2 \times 8.6) {m}^{2} = 17.2 {m}^{2} \)

Answer:

y=5/3x−6

5/3 is your slope

Answer:

see explanation

Step-by-step explanation:

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

Given

3y = 5x - 18 ( divide all terms by 3 )

y = \(\frac{5}{3}\) x - 6 ← in slope- intercept form

with slope m = \(\frac{5}{3}\) and y- intercept c = - 6

Answer: 20cm

Step-by-step explanation:

First, add up the 3 sides you already know.

6+6+3 = 15.

Now you have to find the length of the semi circle which is half the circumference of a circle.

The equation for the circumference is 2πr.

The diameter of the circle is 3 meaning the radius is 1.5.

Plug this into the equation.

2 * π * 1.5 and you get approximately 9.42
Since it is only have a circle, you have to divide this by 2 to get half the circumference.

When you do this, you get 4.71

Add up 15 and 4.71 and you get 19.71 which rounds up to 20.

- Events A, B, C, and D are exhaustive.
- Events A and B, B and D are not mutually exclusive.
- Events A and C, C and D, A and D are mutually exclusive.

To determine whether the events are exhaustive or mutually exclusive, we need to understand the definitions of these terms:
1. Exhaustive events: Events are considered exhaustive if the union of all the events covers the entire sample space S. In other words, there are no outcomes in the sample space that are not included in any of the events.
2. Mutually exclusive events: Events are considered mutually exclusive if they have no outcomes in common. In other words, the events cannot occur simultaneously.
Now let's analyze the given events:
A = {1, 2, 4}
B = {1, 2, 4, 5, 6}
C = {5, 6}
D = {2, 3, 6}
To determine if the events are exhaustive, we need to check if their union covers the entire sample space S.
The union of A, B, C, and D is {1, 2, 3, 4, 5, 6}, which covers the entire sample space S. Therefore, the events A, B, C, and D are exhaustive.

To determine if the events are mutually exclusive, we need to check if any outcomes are shared between the events.
The outcomes 1, 2, and 4 are shared between events A and B. Therefore, events A and B are not mutually exclusive.
The outcomes 2 and 6 are shared between events B and D. Therefore, events B and D are not mutually exclusive.
No outcomes are shared between events A and C, C and D, or A and D. Therefore, events A and C, C and D, and A and D are mutually exclusive.

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The polar curve r = -2 sin θ can be graphed by first plotting the graph of r as a function of θ in Cartesian coordinates. To do this, we can set r = y and θ = x, and then plot the resulting equation y = -2 sin x.

This graph will have the shape of a sinusoidal wave with peaks at y = 2 and troughs at y = -2.
To sketch the same polar curve in Cartesian form, we can use the conversion equations x = r cos θ and y = r sin θ. Substituting in the given polar equation, we get x = -2 sin θ cos θ and y = -2 sin² θ. Simplifying these equations, we get x = -sin 2θ and y = -2/3 (1-cos² θ). This graph will have the shape of a four-petal rose.
The graphs from Part (a) and Part (b) are different because they represent different equations. Part (a) is the graph of y = -2 sin x, which is a sinusoidal wave. Part (b) is the graph of a four-petal rose. However, both graphs share some similarities in terms of their shape and symmetry. They are both symmetrical about the origin and have a repeating pattern.
In conclusion, we can sketch a polar curve by first graphing r as a function of θ in Cartesian coordinates and then converting it to Cartesian form. The resulting graphs may look different, but they often share similar patterns and symmetries.

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Use the appropriate angle rule(s) to solve angles in Mathematical problems.

When two rays are united at a common point, an angle is created. The two rays are referred to as the arms of the angle, while the common point is referred to as the node or vertex. The symbol '∠' stands for the angle. The Latin word "Angulus" is where the term "angle" originated.

These angles are usually measured in degrees like 45°, 30°, 50°, 15°, 110°, 75° and 120° etc.  

Here are some key points and rules to solve angles in Maths

Angles on a straight line

A straight line's angles always add up to 180 degrees. We use this rule to solve angles lying on a straight line.

Angles around a point or angle in a circle

Angles that circle a point always equal 360 degrees. We use this rule to solve angles related to circles, sectors, closed polygons, etc.  

Supplementary angles:

The pair of angles that are added up to 180° are known as Supplementary angles. Example, 150°, 30°;  120°, 60°;  100°, 80° etc.

Complementary angles

The pair of angles that are added up to 90° are known as angles Complementary angle. Examples, 45°, 45°; 50°, 40°; 60°, 40° etc.  

Vertically opposite angles

The angles formed by intersecting two straight lines opposite to each other are known as Vertically opposite angles. In any situation, the vertically opposite angles will equal each other

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Solution

The probability of getting 3 heads from 3 coins is

The expected value will be

The answer is 3 students

The difference quotient for the function is 8.

The function is given by:

f ( x ) = 8 x − 9, where h ≠ 0

To find f(a), substitute a for x in the function. So we have:

f ( a ) = 8 a − 9

To find f(a + h), substitute a + h for x in the function. So we have:

f ( a + h ) = 8 ( a + h ) − 9

The difference quotient can be found using the formula:

(f(a + h) - f(a))/h

Substituting the values found above, we have:

(8 ( a + h ) − 9 - (8 a − 9))/h

Expanding the brackets and simplifying, we have:

((8a + 8h) - 9 - 8a + 9)/h

= 8h/h

= 8

Therefore, the difference quotient for the function is 8.

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The exact value of the expression cos(-10pi/3) is -1/2.

To find the exact value of cos(-10pi/3), follow these steps:

1. Determine the equivalent positive angle: Since the cosine function has a period of 2pi, we can add multiples of 2pi to the angle until we get a positive angle. In this case, we add 4pi (since 4pi = 12pi/3) to get the equivalent positive angle:
  (-10pi/3) + (12pi/3) = 2pi/3.

2. Find the cosine value of the positive angle: Now, we find the cosine value of the positive angle 2pi/3. Using the unit circle, we can determine that cos(2pi/3) = -1/2.

So, the exact value of the expression cos(-10pi/3) is -1/2.

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The measure of ∠EFD as shown in the diagram is 60.5°.

Tangent line to a plane curve at a given point is the straight line that just touches the curve at that point.

Note: The tangent line of a circle and the radius meets at an angle of 90°

To calculate the value of ∠EFD, we use the theorem of the angle susbtends by an arc of a circle.

From the diagram,

First we need to look for ∠DAE

From the diagram,

Substitute these values into equation 2

Substitute the value of ∠DAE into equation 1

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Answer:

0.4

Step-by-step explanation:

1-0.6=0.4???????????????

Answer:

yes

Step-by-step explanation:

The graph represents a function because it passes the vertical line test

From the complete question, the given graph is a linear graph

All linear graphs represent a function

This is so because they have constant slopes, and they pass the vertical line test

Hence, we can conclude that the graph represents a function

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Answer:

y=1/4x+3/2

Step-by-step explanation:

You must first find the slope of the coordinates. (3-1=2)(6-(-2)=8)(2/8 simplifies to 1/4). Then you can use y-y1=m(x-x1) or do y=mx+b. I will show y=mx+b. So, you know the slope now and you also have coordinates so you can plug in -2 for the x and 1 for the y and 1/4 for the m but you leave the b as itself because we don´t know that yet. (1=1/4(-2)+b). You then get (1= -1/2+b) and you want b by itself so you must add 1/2 because it is a negative which you give you (3/2) so now you have you m which is 1/4 and your b which is 3/2 so your equation is (y=1/2x+3/2)

Hope this helped!