Cindy has a rectangular patio that is 20 feet wide and 40 feet long. She plans to expand her patio by increasing both dimensions by a factor of 0.25. Enter the area, in square feet, of Cindy's expanded patio.

The length of the hypotenuse is approximately 50.16.

As we can see in the given right angle triangle that is made in the given model,

the base is 50 and the height is 4, so  for hypotenuse,

Let's label the hypotenuse as 'c.'

We have:

[tex]y^2 = 50^2 + 4^2\\\\y^2 = 2500 + 16\\\\y^2 = 2516[/tex]

To find the value of 'y,' we take the square root of both sides:

y ≈ √(2516)

y ≈ 50.16

For the slope of the given triangle,

In general slope = Δy(horizontal)/Δx(verticle)

The slope = 4/50 = 1/12.5

This slope is under state regulation since it falls between the standard ratio.

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Answer:

---------------------------

Multiply the two numbers first:

Round the number to the first digit:

Write 80000000 as the product of a single digit and a power of 10:

Answer:

630

Step-by-step explanation:

if it takes 1 hour for it to fill up by 70 ft³

then after 9 hours it is full.

9 X 70 = 630 ft³

Thus, the value of s is 8.2 in. we found value of a variable in a given equation or problem by understanding the concept of variables and using algebraic manipulation to isolate the variable.

In order to find the value of s in the given equation, we need to understand the concept of variables. Variables are quantities that can change or vary in a given equation or problem.

In this case, we have three variables: r, s, and t.

The given equation is: r + s + t = 20.7 in.

We are given the values of r and t, which are 2.4 in. and 10.1 in. respectively.

We need to find the value of s.

To find the value of s, we can use algebraic manipulation of the given equation. We can subtract r and t from both sides of the equation to isolate the value of s. This gives us:
s = 20.7 in. - r - t

Substituting the given values of r and t, we get:
s = 20.7 in. - 2.4 in. - 10.1 in.
s = 8.2 in.
Therefore, the value of s is 8.2 in.

In this case, we used the given equation and subtracted the values of the other variables to find the value of s.

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If the "non-wage" income "M" increases, then "Reservation-wage" will also increase.

The "Reservation-Wage" will go up, because if utility function is positive, the reservation wage will increase because the non-wage income increases. But, there has to be some other element which modify the value of the utility function.

The "Reservation-Wage" will be affected if there is an increase in the "non-wage" income M in 2-ways.

Both, the "overall-income" : (U(R+C,M)) and "leisure-time" we have to spend on leisure-related purchases (R+C) will increase.

The "Utility-Function" U(C,R) will also rise by same amount as the "non-wage" income "M". So, "Reserved-Wage" will also rise by same amount.

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The given question is incomplete, the complete question is

Suppose your utility function is given by U(C, R) = ln(R) + C, where R is leisure and C is your aggregate consumption. If your non-wage income "M" increases, how will this affect your reservation wage?

Statement C claims that the domain of h(t) includes all real numbers.

In this context, it is reasonable to assume that time can be any real number since it can vary continuously.

Therefore, Statement C is true.

In the given scenario, the height of a ball thrown vertically upward from the ground is represented by the function[tex]h(t) = 100t - 16t^2,[/tex]

where h(t) represents the height in feet and t represents the time in seconds.

To determine the domain of h(t) in this context, we need to consider the restrictions on the input variable t that make sense within the problem's context.

Statement A claims that the domain of h(t) includes all positive whole numbers.

However, in this context, using positive whole numbers for time would imply that the ball is thrown at discrete moments in time, which may not be the case.

Therefore, Statement A is not true.

Statement B suggests that the domain of h(t) includes all positive integers.

Similar to Statement A, using positive integers for time implies discrete moments, which may not be appropriate for a continuous measurement like time.

Therefore, Statement B is also not true.

Statement C claims that the domain of h(t) includes all real numbers.

In this context, it is reasonable to assume that time can be any real number since it can vary continuously.

Therefore, Statement C is true.

Statement D states that the domain of h(t) includes all non-negative numbers. Since time cannot be negative in this context, this statement is also true.

In conclusion, both Statements C and D are true about the domain of h(t) in this context.

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The complete question may be like:

Question: A ball is thrown vertically upward from the ground. The height of the ball after t seconds is given by the function h(t) = 100t - 16t^2, where h(t) represents the height in feet. Determine which statement is true about the domain of h(t) in this context.

A) The domain of h(t) includes all positive whole numbers.

B) The domain of h(t) includes all positive integers.

C) The domain of h(t) includes all real numbers.

D) The domain of h(t) includes all non-negative numbers.

Sampling is indeed the process of selecting survey respondents or research participants. This statement is true.

Sampling allows researchers to collect data from a smaller, representative group, rather than attempting to gather information from an entire population. This makes the research process more efficient, cost-effective, and manageable. There are various sampling methods, such as random sampling, stratified sampling, and convenience sampling, each with its own advantages and disadvantages depending on the research goals.
A well-designed sampling strategy ensures that the sample accurately reflects the larger population, allowing for generalizable results and meaningful conclusions. It is crucial to consider factors such as sample size and selection bias when designing a research study, as these factors can significantly impact the validity and reliability of the findings. By carefully selecting a representative sample, researchers can increase the likelihood that their results will be applicable to the broader population of interest.
In conclusion, the statement that sampling is the process of selecting survey respondents or research participants is true. This technique is essential in many research scenarios as it enables researchers to gather valuable data and insights from a smaller, manageable group that accurately represents the larger population. Choosing the appropriate sampling method and considering factors such as sample size and selection bias are crucial steps in ensuring the validity and generalizability of the study's findings.

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The first matching scenario is a shortage, the second matching scenario is a surplus, the third matching scenario is a surplus, and the fourth matching scenario is a shortage. The multiple-choice answers are: 1. Shortage, 2. Shortage, 3. Surplus, 4. Surplus.

Matching:

Demand is more than supply - Shortage

Supply is more than demand - Surplus

Oranges at Kind Soopers are priced too high and people don't buy them. As a result, there are oranges that are sitting on the shelves for so long that they are going bad - Surplus

After going viral on social media, demand for Stanley water bottles increases. As a result, the water bottles are very difficult to find - Shortage

Multiple Choice:

II - Shortage

Shortage

Surplus

Surplus

In the first matching scenario, there is a shortage because the demand exceeds the supply. In the second scenario, there is a surplus because the supply is more than the demand. In the third scenario, there is a surplus of oranges because people are not buying them due to high prices, leading to spoilage. In the fourth scenario, there is a shortage of Stanley water bottles due to increased demand after going viral on social media.

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True. If the derivative (dy/dx) of a function f(x) is zero at a particular value of x, then the slope of the tangent line at that point is also zero, which means it is a horizontal line.

A tangent line is a straight line with the same slope as the curve it touches at a single point on a curve. A local approximation of the curve close to the point of contact is provided. Finding the slope of the curve at a given location, which is determined by the derivative of the curve at that position, is necessary to determine the equation of a tangent line to a curve at that point. The equation of the tangent line is then written using the point-slope form of a line. Calculus relies on tangent lines to help students comprehend how functions and their derivatives behave.

This is because the derivative represents the rate of change (slope) of the function at any given point, and if it is zero, then the function is not changing (not curving) at that point.

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The integer that represents how much John has received after two payments is $18. If Kelsey owes John $45 and splits it into five equal payments, each payment would be $9.

After John has received two payments, he would have received a total of $18. However, since the problem is asking for an integer value, we can round down to the nearest dollar.

It is important to note that rounding down to the nearest dollar may not always be appropriate, especially in more complex problems. It is always important to carefully read the question and understand the context before determining the appropriate level of precision to use in the solution.

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The critical value of F for a one-way between-subjects ANOVA with 4 and 43 degrees of freedom (5 levels minus 1, and 48 total participants minus 5 levels) at an alpha level of .01 is approximately 3.737.

To calculate the critical value of F, we need to use a statistical table or calculator. The F distribution table with 4 and 43 degrees of freedom at an alpha level of .01 gives a critical value of 3.737.

This means that if the calculated F value for the ANOVA is greater than 3.737, we can reject the null hypothesis at the .01 level of significance.

It's important to note that the critical value of F changes depending on the degrees of freedom and the alpha level chosen.

In this case, we have 5 levels/conditions and 48 participants, but if the sample size or number of levels changes, the critical value of F would be different.

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(a) The uniform distribution is a symmetric distribution, and therefore the skewness of X is 0.

(b) The variance of X is given by Var[X] = (b-a)^2/12 = (1-(-1))^2/12 = 1/3. Therefore, the standard deviation of X is σ = √(1/3) and the characteristic function of X is given by Øx(t) = E[e^(itX)] = (e^(it) - e^(-it))/(it(b-a)) = (sin(t))/(t).

(c) The expected value of X can be obtained from the first derivative of the characteristic function evaluated at t=0. Therefore, E[X] = Øx'(0) = d/dt(sin(t)/t)|_(t=0) = 1.

The skewness of a continuous random variable X with probability density function f(x) is a measure of the asymmetry of the distribution.

It is defined as the third standardized moment of the distribution, For the uniform distribution on the interval [-1, 1], the mean μ = (1 - (-1)) / 2 = 0 and the standard deviation σ = sqrt((1 - (-1))^2 / 12) = sqrt(1/3).

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Complete Question

Suppose that X ~ unif(-1,1) is a continuous RV. (a) Find skewness of X. (b) Find ox(t). (c) Find E [X] using Øx(t)

Since the limit is less than 1, the taylor series converges for all values of x within a distance of R = 2 units from the center of the series, which is x = 3 in this case. Therefore, the radius of convergence is R = 2.

The Taylor series for f centered at 3 is given by:

f(x) = ∑ [f^(n)(3) / n!] (x - 3)^n

Substituting f^(n)(3) = (-1)^n n! / 2^n (n + 1), we get:

f(x) = ∑ [(-1)^n / (2^n (n + 1))] (x - 3)^n

To find the radius of convergence, we can use the ratio test:

lim┬(n→∞)⁡|(-1)^(n+1) / (2^(n+1) (n+2))| / |(-1)^n / (2^n (n + 1))| = 1/2

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Since we are looking for the range that contains 60% of the values, we need to look at the middle 60% of the distribution.

Thus, we need to find the range that lies within two standard deviations of the mean since this covers 95% of the distribution. To find the range between two x values that contain 60% of the values, we can subtract the range outside of two standard deviations from 100% and divide the result by 2.

This gives us 20%, which means that 10% of the values lie outside of two standard deviations on each end. Since we assume that the distribution is symmetric, we can find the x values by looking at the mean plus and minus two standard deviations.

The area between the mean and two standard deviations is 47.5% (which is half of the remaining 95% after we take out the 2.5% on each end). Therefore, we can estimate that 60% of the values lie between the x values that are 1.96 standard deviations away from the mean on each side.

Using this estimation, we can find the x values by multiplying the standard deviation by 1.96 and adding and subtracting the result from the mean.

Assuming a standard normal distribution (with a mean of 0 and a standard deviation of 1), the x values would be -1.96 and 1.96. If we have a distribution with a known mean and standard deviation, we can use these values to find the corresponding x values.

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Based on the diagram of kite ABCD, each of the angle measure include the following:

10. m∠DAE = 43°.

11. m∠BCE = 55°

12. m∠ABC = 70°.

Based on the diagram of kite ABCD, we can logically deduce that angle ADE and angle DAE would form a complementary angle. This ultimately implies that, the measure of angle DAE can be determined as follows;

m∠DAE + m∠ADE = 90°

m∠DAE = 90° - m∠ADE

m∠DAE = 90° - 47°

m∠DAE = 43°

Question 12

Generally speaking, the sum of the interior angles of a kite is equal to 360 degrees;

m∠ABC + m∠BAD + m∠BCD + m∠BDC + m∠ADE = 360°

m∠ABC + 98 + 98 + 47 + 47 = 360°

m∠ABC + 290 = 360°

m∠ABC = 360° - 290

m∠ABC = 70°

Question 11

m∠BCE = 1/2 × (180° - m∠ABC)

m∠BCE = 1/2 × (180° - 70°)

m∠BCE = 1/2 × (110°)

m∠BCE = 55°

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

The given matrix Q is orthogonal. To see why, note that the dot product of any two columns of Q is equal to zero, which is a necessary condition for a matrix to be orthogonal.

To find the inverse of Q, we can use the fact that for an orthogonal matrix, its inverse is equal to its transpose. Thus,

Q^-1 = Q^T

Therefore, the inverse of Q is

Q^-1 =

[1/sqrt(2)  1/sqrt(2)]

[1/sqrt(2) -1/sqrt(2)]

Note that we could have also used the fact that for a 2x2 orthogonal matrix, its inverse can be found by swapping the elements on the diagonal and changing the sign of the off-diagonal elements. In this case, we have  Q^-1 =

[1/sqrt(2)  1/sqrt(2)]

[1/sqrt(2) -1/sqrt(2)]

which is the same as the result obtained by taking the transpose of Q.

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Therefore, Plugging these values into the formula gives a variance of 3.25.

To find the variance of a binomial experiment, we use the formula:
Variance = n*p*q
Where n is the number of trials, p is the probability of success, and q is the probability of failure (1-p).
In this case, n = 13, p = 0.50, and q = 0.50.
So the variance of the random variable associated with this experiment is:
Variance = 13*0.50*0.50
Variance = 3.25
The variance of a binomial experiment with 13 independent trials and a probability of success of 0.50 is 3.25. This can be calculated using the formula variance = n*p*q, where n is the number of trials, p is the probability of success, and q is the probability of failure (1-p). In this case, the number of trials is 13, the probability of success is 0.50, and the probability of failure is also 0.50.

Therefore, Plugging these values into the formula gives a variance of 3.25.

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a) The polynomial for the area of the soccer field is given as follows: A = x² - 20x - 300.

b) The area when x = 90 is given as follows: A = 6000 yd².

c) The time it takes is given as follows: 90 minutes.

Considering a rectangle of length l and width w, we have that:

The dimensions for this problem are given as follows:

(x + 10) and (x - 30).

Hence the polynomial for the area is obtained as follows:

A = (x + 10)(x - 30)

A = x² - 20x - 300.

When x = 90, the area is given as follows:

A = 90² - 20(90) - 300

A = 6000 yd².

The time it takes to mow the field is given as follows:

6000/200 x 3 = 90 minutes.

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For the number of ways in which six boys and six girls can sit alternately in a line, denoted as x, we can calculate it as x = P(6) * P(6) / (P(6))^6 * 2!, where P(n) represents the permutation of n objects.

To find x, we first arrange the six boys in a line, which can be done in P(6) ways. Next, we arrange the six girls in the 6 spaces between the boys, resulting in P(6) arrangements. However, since the girls can be arranged in any order within each space, we divide by (P(6))^6 to account for duplicate arrangements. Finally, we divide by 2! to consider the two possible arrangements of boys and girls (e.g., boys first or girls first). This gives us the total number of permutations, or ways, in which the boys and girls can sit alternately in a line, which is x.

Similarly, for the circular arrangement, denoted as y, we can calculate it as y = P(5) * P(6) / (P(6))^6 * 2!.

To find y, we first arrange the six boys in a circle, which can be done in P(5) ways (as there are five relative positions for the boys in a circle). Then, we arrange the six girls in the six spaces between the boys, resulting in P(6) arrangements. We divide by (P(6))^6 to account for duplicate arrangements within each space. Finally, we divide by 2! to consider the two possible rotations of the circle. This gives us the total number of permutations, or ways, in which the boys and girls can sit alternately in a circular arrangement, which is y.

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Answer:

Surface area = 1570.8 in.^2

Step-by-step explanation:

The formula for surface area of cylinder is given by:

SA = 2πrh + 2πr^2, where

Since the radius is 10 in. and the height is 15. in, we can find the surface area of the cylinder in square in. by plugging in 10 for r and 15 for h in the surface area formula and simplifying then rounding to the nearest tenth:

SA = 2π(10)(15) + 2π(10)^2

SA = 2π(150) + 2π(100)

SA = 300π + 200π

SA = 500π

SA = 1570.796327

SA = 1570.8 in.^2

Thus, the surface area of the cylinder is about 1570.8 in.^2

The shortest possible length of the line segment cut off by the first quadrant and tangent to the curve y = f(x) is the distance between the origin and the point of tangency.

Consider a curve y = f(x) that passes through the origin and lies entirely in the first quadrant. Let P = (a, f(a)) be a point on the curve where the tangent line at P is parallel to the x-axis. Then, the line segment cut off by the first quadrant and tangent to the curve at P is the line segment from the origin to P.

Since the tangent line at P is parallel to the x-axis, its slope is zero. The slope of the tangent line at P is also equal to the derivative of the curve at a, f'(a). Therefore, we have:

f'(a) = 0

This implies that a is a critical point of the curve, which means that either f'(a) does not exist or f'(a) = 0. Since the curve lies entirely in the first quadrant and passes through the origin, we must have f(0) = 0 and f'(0) > 0.

If f'(a) = 0, then P = (a, f(a)) is a point of inflection of the curve, and the tangent line at P is horizontal. In this case, the line segment from the origin to P is simply the y-coordinate of P, which is f(a).

If f'(a) does not exist, then P is a corner point of the curve, and the tangent line at P is vertical. In this case, the line segment from the origin to P is simply the x-coordinate of P, which is a.

Therefore, the shortest possible length of the line segment cut off by the first quadrant and tangent to the curve at P is given by the distance between the origin and P, which is either f(a) or a, depending on whether P is a point of inflection or a corner point.

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Answer:

  556 cm²

Step-by-step explanation:

You want the surface area of a cylinder of radius 5 cm, height 12 cm, topped with a cone of height 4 cm.

The surface area of the figure will be ...

  surface area = base area + cylinder lateral area + cone lateral area

The base is a circle of radius 5 cm, so its area is ...

  A = πr² = π(5 cm)² = 25π cm²

The lateral area of the cylinder is the product of its circumference and its height:

  A = 2πrh = 2π(5 cm)(12 cm) = 120π cm²

The lateral area of the cone is half the product of the circumference and its slant height. The slant height can be found using the Pythagorean theorem:

  s² = r² + h²

  s = √(5² +4²) = √(25 +16) = √41 . . . . cm (about 6.403 cm)

Then the lateral area of the cone is ...

  LA = πrs

  LA = π(5 cm)(√41 cm) = 5√41π cm² ≈ 32.016π cm²

This brings the total surface area to ...

  surface area = 25π cm² +120π cm² +5√41·π cm² ≈ 556 cm²

The area of the composite solid is about 556 cm².

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Answer:

median

Step-by-step explanation:

49+45=94

94÷2=47

Answer:

The Median is : 36

The Mode is : Undefined

Step-by-step explanation:

    To find the median of a set of numbers, we need to arrange the numbers in order from least to greatest (or greatest to least) and then identify the middle number. If there is an even number of values, we take the average of the two middle numbers.

First, let's arrange the given numbers in ascending order:

13, 16, 31, 41, 45, 49

    We have six numbers in this set, and since there is an even number of values, we need to take the average of the two middle numbers, which are 31 and 41.

Therefore, the median of the given set of numbers is:

(31 + 41)/2 = 72/2 = 36

----------------------------------------------------------------------------------------------------------

    To find the mode of a set of numbers, we look for the number that appears most frequently. In the given set of numbers:

41, 13, 49, 45, 31, 16

    None of the numbers appear more than once, so there is no mode in this set of numbers.

    Therefore, the mode of this set of numbers is undefined or there is no mode.

Answer:

16 units

Step-by-step explanation:

The formula for the area of a trapezoid is:

A = (1/2)h(b1 + b2)

We are given that the bases have lengths of 26 and 30 and the area is 448. Substituting these values into the formula above, we get:

448 = (1/2)h(26 + 30)

448 = (1/2)h(56)

Multiplying both sides by 2/56, we get:

16 = h

Therefore, the height of the trapezoid is 16 units.

Hope this helps you and have a great day!

We can eliminate the negative root. Therefore, the toddler is at rest at t = 2. So the answer is (C) t = 2 only.

To find when the toddler is at rest, we need to find when the velocity of the toddler is zero. We can find the velocity function by taking the derivative of the position function, which gives us:
v(t) = 12t³ - 24t² - 12t + 24
Now we can solve for when the velocity is zero:
0 = 12t³ - 24t² - 12t + 24
0 = 3t³ - 6t² - 3t + 6
0 = t³ - 2t² - t + 2
0 = (t-1)(t²-t-2)
Using the quadratic formula, we can solve for the roots of the quadratic factor:
t² - t - 2 = 0
t = (1 ± sqrt(1 + 8))/2
t = (1 ± 3)/2
t = 2 or t = -1
Since we are given that t > 0, we can eliminate the negative root. Therefore, the toddler is at rest at t = 2. So the answer is (C) t = 2 only.

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Answer:

the second equation (59-31-[]=10)

the answer is 18

X is a normal distribution with mean 4 and standard deviation 8 (since the variance is 64), therefore P(X ≤ –5.2) is approximately 0.1251.

If X follows a normal distribution N(4,64), then it has a mean (μ) of 4 and a variance (σ²) of 64, with a standard deviation (σ) of 8. To find the probability P(X ≤ -5.2), we need to calculate the z-score for -5.2 and use the standard normal distribution table.
Substituting in the values, we get:
z = (-5.2 - 4) / 8
z = -1.15
Now, we can look up the probability of a standard normal distribution with a z-score of -1.15 using a table or calculator, which gives us:
P(Z ≤ -1.15) = 0.1251
Therefore, P(X ≤ –5.2) is approximately 0.1251.

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28.8% percent of those that have a tablet also have a smartphone.

What is the conditional probability?

The chance of an event occurring while taking into account the outcome of an earlier event is known as conditional probability.

It defines the probabilities as follows:

The likelihood that an event B will occur given that an event A occurred is known as P(B|A).

P(A|B) denotes the likelihood that event A will occur after event B has occurred.

P(A) represents the likelihood that event A will occur.

Here, we have

Given: 35% of the children in kindergarten have a tablet, and 24% have a smartphone. given that 42% of those that have a smartphone also have a tablet.

The events for this problem are given as follows:

Event A: has a tablet.

Event B: has a smartphone.

Hence the probabilities are given as follows:

P(A) = 0.35, P(B) = 0.24, P(A|B) = 0.42.

Hence the conditional probability is of:

P(B|A) = 0.42 x 0.24/0.35 = 0.288.

Meaning that the percentage is of:

0.288 x 100% = 28.8%.

Hence, 28.8% percent of those that have a tablet also have a smartphone.

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To find dy/dx, we can use the chain rule,To find d²y/dx², we can use the quotient rule, The curve is concave up where d²y/dx² > 0, and concave down where d²y/dx² < 0. From part (b), we know that d²y/dx² is negative for all values of t, so the curve is concave down everywhere except for at points where dy/dx is undefined.

a) We can find dy/dx by using the chain rule:

dy/dx = dy/dt ÷ dx/dt

dx/dt = -9cos^8(t)sin(t)

dy/dt = 16sin(t)cos(t)

Therefore,

dy/dx = (16sin(t)cos(t)) / (-9cos^8(t)sin(t))

= -16cos(t) / (9sin^2(t)cos^7(t))

= -16cot(t) / (9cos^6(t))

(b) To find d²y/dx², we differentiate dy/dx with respect to t:

d(dy/dx) / dt = d/dt (-16cot(t) / (9cos^6(t)))

= (16 / 9) csc^2(t) cot^2(t) - (96 / 9) cos^5(t) csc^2(t) cot(t)

= (16 / 9) csc^2(t) (cot^2(t) - 6cos^5(t) cot(t))

Now, using the fact that dy/dx = -16cot(t) / (9cos^6(t)), we can write

d²y/dx² = (d(dy/dx) / dt) ÷ (dx/dt)

= [(16 / 9) csc^2(t) (cot^2(t) - 6cos^5(t) cot(t))] ÷ [-9cos^8(t)sin(t)]

= -16csc^2(t) / (9cos^7(t)) + (96 / 9) csc^2(t) cos^4(t) / sin(t)

= (16 / 9) csc^2(t) (6cos^4(t) / sin(t) - cot^2(t) - 1 / cos^7(t))

(c) To determine the concavity of the curve, we look at the sign of d²y/dx². If d²y/dx² is positive, the curve is concave up. If d²y/dx² is negative, the curve is concave down.

Note that dy/dx is undefined at t = kπ, where k is an integer, because cos^6(t) = 0. However, these points do not affect the concavity of the curve.

We can simplify d²y/dx² as

d²y/dx² = (16 / 9) csc^2(t) [(6cos^4(t) / sin(t)) - cot^2(t) - 1 / cos^7(t)]

The expression inside the square brackets is always positive, since cos^4(t) and cos^7(t) are both positive for all t and 1/sin(t) is positive for 0 < t < π. Therefore, the sign of d²y/dx² is determined by the factor csc^2(t), which is positive for 0 < t < π/2 and π/2 < t < π, and negative for π < t < 3π/2 and 3π/2 < t < 2π.

Therefore, the curve is concave up for 0 < t < π/2 and π/2 < t < π, and concave down for π < t < 3π/2 and 3π/2 < t < 2π.

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The solution is, the root of the equation is: (x, y) = (8, 5)

Here, we have,

The equations are:

−5x+8y=0 ...1

−7x−8y=−96 ....2

Adding (1) and (2), we get:

-12x = -96

or, x = 8

⇒ x = 8

Substituting x = 8, in Equation (1), we get:

8y = 5x

8y = 40

⇒ y = 5

Therefore, the root of the equation: (x, y) = (8, 5).

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