Graph each pair of inequalities below and indicate the solution set of the system with crosshatching or shading. The crosshatching or shading, if extended, would cover a set of three letters. Print these letters in the three boxes at the bottom of the page that contain the exercise number.

The new price for the arug is $87.14 when Astore marks up the price by 7%.

The new price for the arug can be calculated by using the following formula:

New Price = (Original Price) + (Original Price) x (Markup Percentage)

In this case, the original price of the arug is $82.00 and the markup percentage is 7%. Therefore, the new price would be:

New Price = $82.00 + ($82.00 x 0.07)

New Price = $87.14

Therefore, the new price for the arug is $87.14 when Astore marks up the price by 7%.The calculation can be further explained by looking at the markup percentage. This percentage is the difference between the original price and the new price, divided by the original price. In this case, the difference between the original price and the new price is $5.14, and when this is divided by the original price of $82.00, the result is 0.0627 (or 6.27%). This is the same as the markup percentage of 7%.

This calculation is useful for any situation where the original price and the markup percentage are known. The new price can be calculated by adding the original price and the amount determined by the markup percentage.

Question :

Astore pays $82.00 for arug and marks the price up by 7%. What is the price of arug after Astore marks it up by 7%?

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The required nth terms of the different arithmetic sequence are as follow,

A15 = 74 for A1= 4 , d = 5

A20 = 90 , for A1 = 14 , d =4

A18 = -23 for A1 = 45 , d = -4

For an arithmetic sequence with first term,

A1 = 4 and common difference d = 5,

The formula for the nth term is equal to,

An = A1 + (n - 1)d

Substituting the given values, we have,

⇒ A15 = 4 + (15 - 1)5

⇒ A15 = 4 + 70

⇒ A15 = 74

The 15th term of the arithmetic sequence is 74.

Now for,

first term A1 = 14 and common difference d = 4,

Substituting the given values, we have,

⇒ A20 = 14 + (20 - 1)4

⇒ A20 = 14 + 76

⇒ A20 = 90

The 20th term of the arithmetic sequence is 90.

For first term A1 = 45 and common difference d = -4,

Substituting the given values, we have,

⇒ A18 = 45 + (18 - 1)(-4)

⇒ A18 = 45 - 68

⇒ A18 = -23

The 18th term of the arithmetic sequence is -23.

Therefore, for the respective arithmetic sequence the required terms are as follow,

A15 = 74, A20 = 90 , and A18 = -23

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The side length of the cube from the volume is s = ∛729

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

V(s) = s³

Where

V(s) = volume

s = side length

From the question, we have

V = 729

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

s³ = 729

Take the cube roots

s = ∛729

Hence, the expression of side length is s = ∛729

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The value of the between groups mean square that would be reported in the ANOVA test is 7.553 and the value of the within groups mean square that would be reported in the ANOVA test is 466.187.

In ANOVA, the "between groups" mean square is calculated by dividing the sum of squares between groups by the degrees of freedom between groups. The "within groups" mean square is calculated by dividing the sum of squares within groups by the degrees of freedom within groups.

These values help to determine whether there is a significant difference in means between the groups. In this case, the between groups mean square is relatively small compared to the within groups mean square, indicating that there is not a significant difference in mean daily sales among the three locations.

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The exponential model for the present population is:

[tex]P = 117,000(1+0.03} )^t[/tex] and the population in 2020 is 152,658.

What is an exponential function?

A mathematical function called an exponential function is employed frequently in everyday life. It is mostly used to compute investments, model populations, determine exponential decline or exponential growth, and so forth. When the input variable x appears as an exponent in the formula f(x) = aˣ, an exponential function is indicated. The exponential curve is influenced by both the exponential function and the value of x. A constant serves as the exponent in an exponential function, but not the other way around (if a function has a variable as the base and a constant as the exponent then it is a power function but not an exponential function).

The population in 2011 = 117,000

Rate of Increase in population each year = 3%

We are asked to find the exponential model for population y of the city, t years after 2011.

Initial population a = 117,000

r = 3%

Present population P = [tex]a(1+\frac{r}{100} )^t[/tex] = [tex]117,000(1+0.03} )^t[/tex]

Now we are asked to find the population in 2020.

t = 2020 - 2011 = 9

P = [tex]117,000(1+0.03} )^t = 117,000 (1.03)^9[/tex] = 152,658.46 = 152,658.

Therefore the exponential model for the present population is:

[tex]P = 117,000(1+0.03} )^t[/tex] and the population in 2020 is 152,658.

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An equation that models the situation with r, the cost of 1 rake: 12r = 204

Here, r represents the cost of a rake.

A gardener buys a rake and a pair of gloves for each of the 12 volunteers in the gardening club.

So, using unitary method the cost of 12 rakes would be: 12 × r = 12r dollars

Each pair of gloves cost $8. 0

So, using unitary method the cost of 12 pair of gloves would be:

12 × 8 = 96 dollars

The gardener spends a total of $300 on both rakes and gloves.

So, we get an equation:

12r + 96 = 300

12r + 96 - 96 = 300 - 96

12r = 204

Therefore, the required equation is: 12r = 204

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The probability that more than the expected number of Sikh youths wear a turban in a random sample of five is 0.3671 (rounded to 4 decimal places).

a. Using the binomial distribution formula, we can determine the likelihood that two out of a random sample of five Sikh teenagers are turban-wearing:

[tex]P(X = 2) = (5 pick 2) (5 choose 2) * (0.75)^3 * (0.25)^2[/tex]

"X" indicates the proportion of Sikh adolescents who wear turbans, "5 choose 2" indicates the number of possible methods to select 2 Sikhs from a group of 5, and "0.75" and "0.25" indicate the likelihoods that a Sikh youngster will not be wearing a turban and will be wearing one, respectively.

This expression can be made simpler by using a calculator to become:

P(X = 2) = 10 * 0.421875 * 0.0625 = 0.2659

Hence, 0.2659 percent chance exists that two out of every five Sikh youngsters in a random sample will be turban-wearing (rounded to 4 decimal places).

Using the complement rule, we can determine the likelihood that two or more of a random sample of five Sikh teenagers are wearing turbans:

P(X >= 2) = 1 - P(X 2)

where X is the proportion of young Sikhs who are turban-wearing.

P(X 2) is the likelihood that fewer than 2 of five Sikh teenagers chosen at random will be turban-wearing.

This can be calculated as:

P(X 2) equals P(X = 0) plus P(X = 1).

P(X = 0) = (5 select 0) * (0.75) * (5 select 0) * (0.25) * 0 = 0.2373

P(X = 1) = (5 select 1) * (0.75) * (4 select 1) * (0.25) * 1 = 0.3956

P(X 2) = 0.2373 + 0.3956 = 0.6329 as a result.

We can then compute P(X >= 2) = 1 - 0.6329 = 0.3671 using the complement rule.

Consequently, there is a 0.3671 percent chance that two or more of five Sikh teenagers chosen at random will be turban-wearing (rounded to 4 decimal places).

c. The following formula can be used to determine how many Sikh teenagers in a random sample of five are likely to wear a turban:

E(X) = n * p = 5 * 0.25 = 1.25

Using the cumulative binomial distribution function, we can determine the likelihood that more Sikh teenagers than predicted in a random sample of five wear turbans:

P(X > 1) = 1 - P(X <= 1)

P(X = 1) equals P(X = 0) plus P(X = 1).

P(X = 0) = (5 select 0) * (0.75) * (5 select 0) * (0.25) * 0 = 0.2373

P(X = 1) = (5 select 1) * (0.75) * (4 select 1) * (0.25) * 1 = 0.3956

Therefore,

P(X <= 1) = 0.2373 + 0.3956 = 0.6329

The complement rule can then be used to determine:

P(X > 1)

P(X > 1) = 1 - 0.6329 = 0.3671

Hence, in a random sample of five Sikh adolescents, the likelihood that there are more turban-wearing Sikh youths than expected is 0.3671. (rounded to 4 decimal places).

d. To determine the likelihood that more than predicted will occur.

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The statement for a projectile motion which is a curvilinear motion, the proper coordinate system to be used for analysis should not be a rectangular system like x-y is False.

Projectile motion is a type of motion in which an object is thrown near the earth's surface, and it moves along a curved path under the action of gravity. The path followed by a projectile motion is called a trajectory.

The trajectory of a projectile motion can be analyzed using a rectangular coordinate system like x-y. In this system, the horizontal and vertical components of the motion can be analyzed separately. The horizontal component of the motion is constant, while the vertical component is affected by gravity.

Therefore, a rectangular coordinate system like x-y is the proper coordinate system to be used for the analysis of projectile motion.

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Answer:

i already solved the step by step explanation, ask someone else to draw the graph for u

Step-by-step explanation:

• y = cos(Θ) + sin(Θ)

Prediction: The graph of this equation will be a sinusoidal wave with an amplitude of √2 and a period of 2π. It will intersect the y-axis at y = √2 and have a minimum value of -√2 at Θ = 3π/4 and a maximum value of √2 at Θ = π/4.

Graph:

Graph of y=cos(Θ) + sin(Θ)

The graph confirms our prediction.

• y = cos2(Θ)

Prediction: The graph of this equation will be a cosine wave squared, which means it will have the same period as the cosine function but its amplitude will be squared. It will intersect the y-axis at y = 1 and have a minimum value of 0 at Θ = π/2 and Θ = 3π/2.

Graph:

Graph of y=cos2(Θ)

The graph confirms our prediction.

• y = sin2(Θ)

Prediction: The graph of this equation will be a sine wave squared, which means it will have the same period as the sine function but its amplitude will be squared. It will intersect the y-axis at y = 0 and have a minimum value of -1 at Θ = π/2 and a maximum value of 1 at Θ = 3π/2.

Graph:

Graph of y=sin2(Θ)

The graph confirms our prediction.

• y = cos2(Θ) + sin2(Θ)

Prediction: This equation represents the identity cos2(Θ) + sin2(Θ) = 1, so the graph of this equation will simply be a horizontal line at y = 1.

Graph:

Graph of y=cos2(Θ) + sin2(Θ)

The graph confirms our prediction.

This can be seen as an obvious result, as it is the probability of the event occuring within a certain range of trials.

To answer your question, it is first important to understand what is meant by the term "Geometric Random Variable". A geometric random variable is a random variable with a probability distribution that follows a geometric pattern.

In other words, it is a discrete random variable that measures the number of trials needed for a certain event to occur.

In this case, the random variable is x and n is the number of trials. The statement p(x > nk - 1|x > n - 1) can be interpreted as the probability that x is greater than nk - 1, given that x is already greater than n - 1.

In light of the construction of a geometric distribution from a sequence of independent Bernoulli trials, this statement can be interpreted as the probability that x is greater than the number of trials multiplied by k, given that x is already greater than the number of trials minus one.

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Algorithm to enumerate all permutations of numbers {1, 2, ..., n} using an explicit stack:

1: Initialize an array s with n elements.

2: Create an empty stack, call it stack.

3: Initialize a counter variable, called i, to 1.

4: Push i onto the stack and increment i by 1.

A stack is used to store the elements, and the procedure keeps track of the permutation that has been created so far by the stack. If there are still elements left to add, the algorithm will add them to the stack in order, one by one. If the stack is full, it will be printed, and the top element will be removed.

The counter variable, i, is used to keep track of the current element being added to the stack.

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Answer:

Step-by-step explanation:

Can you explain a bit further of this question, so then I can help you?

Answer:

If Jamil spent 2/5 of his money on a box of chocolates, then the fraction of money that is left with him is 1/5.

Let the amount that Jamil started with be = $x,

Jamil spent 2/5 of his money on a box of chocolates, which means he spent (2/5)x dollars.

So, He now has (3/5)x dollars remaining.

Next, Jamil spent (2/3) of the remaining money on cupcakes, which means he spent (2/3)(3/5)x = (2/5)x dollars on cupcakes.

He now has : x - {(2/5)x + (2/5)x} = (1/5)x dollars left.

Therefore, Jamil had 1/5 of his money left.

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Therefore , the solution of the given problem of function comes out to be there is a 0.061071 absolute difference between the integral's approximate value and its real value.

There will be questions on design, arithmetic, each subject, and both real and imagined locations on the midterm exam. An account of the relationships between different elements that combine to create the same result. A service is composed of numerous distinctive components that work in tandem to produce distinctive results for each input.

Here,

We must divide the interval [1, 0] into two subintervals of equal length in order to use a midway Riemann sum with two intervals of equal length:

=> [1, 1/2] and [1/2, 0]

Each subinterval's breadth is given by x = (1-0)/2 = 1/2, and its midpoint is given by

=>x1 = (1 + 1/2)/2 = 3/4 and x2 = (1/2 + 0)/2 = 1/4

Therefore, the midway Riemann sum is:

=> F(x1,x) + F(x2,x)

=>  (3/4)^2 * e^(-3/4) * 1/2 + (1/4)^2 * e^(-1/4) * 1/2

=> 0.099532

The exact discrepancy between this estimate and the true integral value is:

=> |0.160603 - 0.099532| = 0.061071

As a result, there is a 0.061071 absolute difference between the integral's approximate value and its real value.

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The unit tangent vector T and the principal unit normal vector N for the parameterized curve is zero

Let's consider two parameterized curves, r(t) = <10 cos t², 10 sin t²> and r(t) = <cos t, sin t>. To find the unit tangent vector T and the principal unit normal vector N for the first curve, we first need to compute its derivative. Using the chain rule, we have:

r'(t) = <-20t sin t², 20t cos t²>

The magnitude of r'(t) is given by:

||r'(t)|| = sqrt((-20t sin t²)² + (20t cos t²)²) = 20t

To obtain the unit tangent vector T, we divide r'(t) by its magnitude:

T(t) = r'(t)/||r'(t)|| = <-sin t², cos t²>

Note that T(t) is a vector of magnitude 1, which means that it represents the direction of motion of the curve at the point r(t).

To find the principal unit normal vector N, we need to compute the derivative of T(t):

T'(t) = <-2t cos t², -2t sin t²>

The magnitude of T'(t) is given by:

||T'(t)|| = sqrt((-2t cos t²)² + (-2t sin t²)²) = 2t

To obtain N(t), we divide T'(t) by its magnitude and flip the direction:

N(t) = -T'(t)/||T'(t)|| = <cos t², sin t²>

Note that N(t) is also a vector of magnitude 1, and it is perpendicular to T(t), which means that it represents the direction of curvature of the curve at the point r(t).

Finally, we can verify that T(t) and N(t) are orthogonal by computing their dot product:

T(t) . N(t) = (-sin t²)(cos t²) + (cos t²)(sin t²) = 0

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Option B.  According to the Inscribed Angle Theorem, the measure of angle RNQ is half the measure of arc PR.

The Inscribed Angle Theorem states that an angle formed by two chords in a circle is half the measure of the arc intercepted by the angle.

In the given diagram, angle RNQ is an inscribed angle. It is formed by the two chords RP and NQ which intersect at point P.

Arc PR is the arc intercepted by angle RNQ. It is the part of the circle between points P and R.

According to the Inscribed Angle Theorem, the measure of angle RNQ is half the measure of arc PR. So we have:

angle RNQ = 1/2(arc PR)

This means that option B, "angle RNQ is half the measure of arc PR", is the correct answer.

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How does the measure of angle rnq relate to the measure of arc pr?

A. angle rnq is equal in measure to arc pr.

B. angle rnq is half the measure of arc pr.

C. angle rnq is twice the measure of arc pr.

D. angle rnq is four times the measure of arc pr.

The lateral area of the triangular prism is 378 square inches, and the total surface area is 504 square inches.

A triangular prism is a three-dimensional geometric shape that has two identical parallel triangular bases and three rectangular faces. The triangular prism can be formed by connecting the corresponding vertices of two congruent triangles and three rectangular faces. The rectangular faces are perpendicular to the triangular bases and form the sides of the prism. The height of the triangular prism is the distance between the two triangular bases. The volume of a triangular prism can be calculated by multiplying the area of the base by the height, and the surface area of a triangular prism can be calculated by finding the sum of the areas of the two triangular bases and the three rectangular faces. Triangular prisms are commonly used in architecture, engineering, and mathematics.

To find the lateral area of a triangular prism, you need to calculate the area of all three rectangular faces and then add them together.

First, let's find the area of the triangular base:

Area of triangle = (1/2) x base x height

= (1/2) x 14 x 9

= 63 square inches

The perimeter of the triangular base is simply the sum of the three sides, which is 14 + 14 + 14 = 42 inches.

Now we can find the area of each rectangular face:

Area of rectangular face = length x height

= 14 x 9

= 126 square inches

There are three rectangular faces, so the total lateral area is:

Lateral area = 3 x area of rectangular face

= 3 x 126

= 378 square inches

To find the total surface area of the prism, we also need to include the two triangular faces. Each triangular face has an area of 63 square inches (as calculated earlier). So the total surface area is:

Total surface area = Lateral area + 2 x area of triangular face

= 378 + 2 x 63

= 504 square inches

Therefore, the lateral area of the triangular prism is 378 square inches, and the total surface area is 504 square inches.

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If the position function of a particle is x-5.0063-4.00t+ 300, with x in meters and t in seconds, then the acceleration of the particle at t = -2.00 s will be 4.00 m/s².

Acceleration of the particle at t = -2.00 s will be 4.00 m/s².

What is the position function of a particle?

The position function of a particle gives the position of a particle as a function of time, t.

In other words, it is the function that specifies the location of a particle relative to a reference point or origin as a function of time.

The position function of a particle given is: x = -5.0063 - 4.00t + 300 where x is in meters and t is in seconds.

How to find acceleration of the particle?

We know that acceleration is the rate at which velocity changes with time, and velocity is the rate at which position changes with time.

In other words,a(t) = d[v(t)]/dt,

where v(t) is the velocity function and a(t) is the acceleration function.

If x(t) is the position function, then we can find the velocity function by differentiating the position function with respect to time, t.v(t) = dx(t)/dt

Therefore, v(t) = d/dt (-5.0063 - 4.00t + 300) = -4.00 m/s

Similarly,

to find the acceleration function, we need to differentiate the velocity function with respect to time. a(t) = dv(t)/dt

Therefore, a(t) = d/dt (-4.00) = 0

Since acceleration is the derivative of the velocity function and velocity is constant, the acceleration is 0 m/s².

Hence, the acceleration of the particle at any time t is 0 m/s².

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Therefore , the solution of the given problem of unitary comes out to be  no eggs bought  307 , 994  and Total 760 , 1240 .

The job can be completed by integrating the information obtained and use this variable method including all supplementary data from two persons who used a specific tactic. This means that, is if desired outcome materialises, either the equation stated entity will be acknowledged or, in actuality, both important processes will skip the colour. For forty pencils, a refundable charge of Rupees ($1.01) might be required.

Here,

The proportion of shoppers who purchased milk and eggs separately should exist if there is no correlation between the factors.

Considering that the table is meant to display the amount of consumers in each category:

                     bought milk      no milk bought

bought eggs           453                246

no eggs bought        307                994

To fill inside the cell for "bought eggs, bought milk", we could perhaps multiply the overall number of shoppers through this proportion: 1000 * 0.453 = 453. In a similar manner, we can complete the remaining compartments using the determined proportions:

                     bought milk      no milk bought

bought eggs           453                246

no eggs bought        307                994

        Total                  760                1240

Keep in mind that the sums for each row and column should correspond to the actual data.

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The area of a circle can be calculated using the formula A = πr^2, where r is the radius of the circle. In this case, the circumference of the circle is 12.56 units, which means that the radius is half of this, or  1.99 units which is approximately 3.90 units .

Step 1: Determine the circle's radius.

The formula for circumference is C = 2πr, where r is the radius of the circle.

2πr = 12.56

r = 12.56 / (2π)

r = 1.99 units

Step 2: Calculate the area of the circle

The formula for area of a circle is A = πr2

A = π(1.99)2

A = 3.90 units2

The area of a circle can be calculated by using the formula A = π[tex]r^2,[/tex] where r is the radius of the circle. The radius of a circle is half of its circumference, so if the circumference of a circle is 12.56 units, the radius is 6.28 units. By plugging this value into the formula, the area of the circle can be found. To calculate the area of the circle, we would multiply π (3.141592654) by [tex]6.28^2[/tex], which would give us 122.58 units. This is the approximate area of the circle with a circumference of 12.56 units. It is important to note that the value of π is an approximate value, and can be rounded up or down depending on the precision desired. Additionally, the value of π can be substituted with other values such as 22/7 or 3.14 to obtain a more accurate result.

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525 radios must be produced and sold for the company to break even.

What is break even?

Break-even is the point of balance where there is neither a profit nor a loss, also known as point of equilibrium. Every value below the break-even mark represents a loss, whereas any value above it represents a gain.

We are given the revenue function as R(x) = 48x and the cost function as C(x) = 15,750 + 18x .

We know that at break even R (x)  = C (x).

So, from this we get

⇒48x = 15,750 + 18x

⇒30x = 15,750

⇒x = 525

Hence, 525 radios must be produced and sold for the company to break even.

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The complete question has been attached below

That's what the genuine statement is (d) both f(x) and g(x) increase on the interval of (0, ∞)

How to decide on a genuine statement?

The functions are given as:

f(x) = log₂(2x)

g(x) = 2x - 3

Then, we plot the diagram of the two functions (see the connection)

From the joined diagram, we can see that:

As x increases, the function's esteem increase

Subsequently, that's what the genuine statement is (d) both f(x) and g(x) increase on the interval of (0, ∞)

Increasing and decreasing intervals of real numbers are the real-esteemed capabilities that will generally increment and reduce with the adjustment of the worth of the reliant variable of the capability.

To find intervals of increment and reduction, you want to decide the primary subordinate of the capability. This is finished to find the indication of the capability, whether negative or positive.

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Answer:

I believe it is 24

Step-by-step explanation:

The area of the figure is 203 square km. The term" area" refers to the  quantum of space  enthralled by a 2D form or  face. We apply cm2 or m2 as our units for measuring area. A shape's area is  refereed by breaking up its length by its breadth.

We have to determine

What is the area of this figure?

How to determine the area of the figure?

The area of the complex figure is determined in the three steps than adding all areas to get the area of the hole figure.

The area of the given figure is determined by the following formula;

The width of the first part is 3 km and the length is 4km.

The width of the second part is = 3+ 9 = 12 km

And the length of the second part is 6 km.

The width of the third part is = 3+ 9 + 5 = 17 km

And the length of the third part is 7 km.

Therefore,

The area of the given figure is;

= 12 + 72 + 117 = 203

Hence, the area of the figure is 203 square km.

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Complete Question: There are 11 players on a soccer squad. Among them, there are 2 attackers, 4 midfielders, 4 defenders, and 1 goalkeeper. 11 soccer players on the field. How many of the players on the pitch are midfielders?

There are 4 midfielders playing in the game in soccer team using substitution in mathematics.

In a soccer team, there are different positions with varying roles, such as attackers, midfielders, defenders, and goalkeepers. In this particular squad, there are 11 players, with 2 attackers, 4 midfielders, 4 defenders, and 1 goalkeeper. To determine how many midfielders are on the field, we need to know the number of players on the pitch.

In soccer, there are 11 players on the field at a time. So, if we consider all the players in the squad, including substitutes, we need to subtract the number of substitutes from the total squad to determine the number of players on the pitch. Typically, there are 3 substitutes in soccer, but this can vary depending on the league or tournament rules.

Assuming there are 3 substitutes, the number of players on the pitch is 11 - 3 = 8.

Since there are 4 midfielders in the squad, and 8 players on the pitch, we can conclude that there are 4 midfielders playing in the game. Midfielders are essential players in soccer, responsible for linking defense and attack, controlling the game's tempo, and creating scoring opportunities. Therefore, having a balanced and competent midfield is critical to a team's success in soccer.

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a) The general solution is y = t^4/4 + t^2/2 + A, b) The general solution is y = t^5/20 + t^3/6 + At + B, c) the general solution is x = t^6/120 + t^4/24 + At^2/2 + Bt + C   d) The general solution is y = 3t^2/4 + t + A/2

(a) To find the general solution of y′=4t^3+t, we need to integrate both sides of the equation with respect to t. This gives us:
y = ∫(4t^3+t)dt = t^4/4 + t^2/2 + A
So the general solution is y = t^4/4 + t^2/2 + A, where A is the first constant of integration.

(b) To find the general solution of y′′=4t^3+t, we need to integrate both sides of the equation twice with respect to t. This gives us:

y′ = ∫(4t^3+t)dt = t^4/4 + t^2/2 + A
y = ∫(t^4/4 + t^2/2 + A)dt = t^5/20 + t^3/6 + At + B

So the general solution is y = t^5/20 + t^3/6 + At + B, where A is the first constant of integration and B is the second constant of integration.

(c) To find the general solution of d^3x/dt^3=4t^3+t, we need to integrate both sides of the equation three times with respect to t. This gives us:
d^2x/dt^2 = ∫(4t^3+t)dt = t^4/4 + t^2/2 + A
dx/dt = ∫(t^4/4 + t^2/2 + A)dt = t^5/20 + t^3/6 + At + B
x = ∫(t^5/20 + t^3/6 + At + B)dt = t^6/120 + t^4/24 + At^2/2 + Bt + C

So the general solution is x = t^6/120 + t^4/24 + At^2/2 + Bt + C, where A is the first constant of integration, B is the second constant of integration, and C is the third constant of integration.

(d) To find the general solution of 2y′=3t+2, we need to integrate both sides of the equation with respect to t. This gives us:

2y = ∫(3t+2)dt = 3t^2/2 + 2t + A
y = (3t^2/2 + 2t + A)/2 = 3t^2/4 + t + A/2

So the general solution is y = 3t^2/4 + t + A/2, where A is the first constant of integration.

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The two line segments are drawn horizontally, and the two rays are drawn vertically intersecting both line segments.

To draw two line segments that are parallel to each other, start by drawing a line segment with a ruler and a pencil. This line segment will serve as the reference line. Draw a second line segment that is the same length and parallel to the reference line. Make sure to use a ruler to ensure that the two lines are exactly parallel. To draw two rays that are perpendicular to both line segments, start by drawing a line segment that is perpendicular to the reference line. This line segment will serve as the reference line for the rays. Draw two separate rays, one at each end of the reference line, so that each ray is perpendicular to both line segments. Make sure to use a ruler to ensure that the two rays are exactly perpendicular.

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The population increases by 15% means if we take the population of deer in the year 2011 as 'x,' then in 2012,

the deer population will be

x + (15/100) x or 1.15x.

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