The foci of an ellipse are (3, 0) and (-3, 0), and the vertices are (7, 0) and (-7, 0) Find an equation of the ellipse. Then sketch the conic section and bring the sketch to your discussion section. Which form (below) does the equation of the given fit? X^2/c^2 + y/d = 1 x/c + y^2/d^2 = 1 x^2/x^2 + y^2/d^2 = 1 x^2/c^2 - y^2/d^2 = 1 y^2/sigma^2 - x^2/c^2 = 1

The probability that they are in middle school is 0.067.

We are given that;

The table of middle school, high school and total.

Now,

The probability that a student selected at random is a middle school student who was present is calculated by dividing the number of middle school students present by the total number of students present.

1,276​/8,632≈0.148

The probability that a high school student selected at random was absent on that day is calculated by dividing the number of high school students absent by the total number of high school students.

12,118​/12,68≈0.167

The probability that a student who was present that day is in middle school is calculated by dividing the number of middle school students present by the total number of students present.

1,276/19,195​≈0.067

Therefore, by probability the answer will be 0.067

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The 90% confidence interval for the mean weekly time spent on homework for the population is (11.6, 12.6).

(a) Find the sample mean and sample standard deviation: The sample mean is calculated by summing up all the values and dividing by the sample size. In this case, the sum of the values is 201.9, so the sample mean is 201.9/16 = 12.61875 hours. The sample standard deviation can be found using the sample computational formula, which calculates the square root of the sum of squared differences between each value and the sample mean, divided by (n-1). After calculations, the sample standard deviation is approximately 1.9784. (b) Find the critical value using the t-distribution: Since the sample size is small (n = 16), we use the t-distribution to find the critical value. With a 90% confidence level, the alpha value is (1-0.90) / 2 = 0.05. Using the t-distribution table or calculator with 15 degrees of freedom (n-1), the critical value is approximately 1.7531. (c) Estimate the sample mean: The sample mean calculated in step (a) represents an estimate of the population mean. Using the formula for the confidence interval: sample mean ± (critical value * (sample standard deviation / sqrt(n))), we can substitute the values into the formula to calculate the confidence interval. In summary, the 90% confidence interval for the mean weekly time spent on homework for the population is calculated using the sample mean, sample standard deviation, critical value from the t-distribution, and the sample size.

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The percent change between the maximum population and the current population for this herd of caribou is -60%. This means that the current population is 60% lower than the maximum population due to habitat loss.

To determine the percent change between the maximum population and the current population for this herd of caribou, we need to use the formula for percent change, which is:
Percent Change = (New Value - Old Value) / Old Value x 100%

In this case, the old value is the maximum population size of 500,000, and the new value is the current population size of 198,000. So, we can plug these values into the formula:

Percent Change = (198,000 - 500,000) / 500,000 x 100%
When we simplify this equation, we get:
Percent Change = -302,000 / 500,000 x 100%

The negative sign indicates a decrease in population, which makes sense given that the current population is lower than the maximum population due to habitat loss. To find the actual percent change, we can divide -302,000 by 500,000 and then multiply by 100:

Percent Change = -0.604 x 100%
Rounding to the nearest whole number, we get:
Percent Change = -60%

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To find the equilibrium volume v for a gas that obeys condition (a) and has an excluded volume, we'll use the given entropy formula s = nkln(v−nb) along with the given value for b. In order to determine the equilibrium volume, we need to find the point where the Gibbs free energy G is minimized. The Gibbs free energy is given by:

G = U + PV - TS
Where U is internal energy, P is pressure, V is volume, T is temperature, and S is entropy. In this case, we are not given values for U, P, or T. Therefore, we cannot explicitly calculate the equilibrium volume v.

However, if more information about the system's temperature and pressure is provided, you could use the equation of state for the gas, along with the given entropy formula, to minimize the Gibbs free energy and find the equilibrium volume v.

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The relative abundance of the (M+2) peak to the M peak for C10H6Br2 is approximately 0.0266%.

To calculate the relative abundance of the (M+2) peak to the M peak for C10H6Br2, we need to consider the natural abundance of carbon and bromine isotopes.

The formula for C10H6Br2 suggests that we have a total of 10 carbon atoms, 6 hydrogen atoms, and 2 bromine atoms. We'll consider the isotopic composition for carbon and bromine.

For carbon isotopes, the natural abundance of 12C is approximately 98.93%, and the natural abundance of 13C is approximately 1.07%.

For bromine isotopes, the natural abundance of 79Br is approximately 50.69%, and the natural abundance of 81Br is approximately 49.31%.

To calculate the relative abundance of the (M+2) peak to the M peak, we need to consider the isotopic contributions of each atom.

For carbon, we have 10 carbon atoms, so the probability of having a (M+2) peak due to one carbon atom is (1.07%)(98.93%)^9(1.07%) = 0.00106432.

For bromine, we have 2 bromine atoms, so the probability of having a (M+2) peak due to one bromine atom is (49.31%)(50.69%) = 0.25006439.

The overall relative abundance of the (M+2) peak to the M peak is the product of the individual probabilities:

Relative abundance = (0.00106432)(0.25006439) ≈ 0.0002662 or approximately 0.0266%.

Therefore, the relative abundance of the (M+2) peak to the M peak for C10H6Br2 is approximately 0.0266%.

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To amortize a loan of $256,400 with an interest rate of 7.2% over 30 years, the monthly house payment can be calculated using the amortization formula. The payment size is $1,758.70 (rounded to the nearest cent).

To find the monthly house payment necessary to amortize the loan, we can use the formula for calculating the monthly payment amount for an amortizing loan. The formula is given by:

Payment = (Loan Amount * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate)^(-Number of Payments))

First, we need to convert the annual interest rate to a monthly interest rate. The monthly interest rate can be calculated by dividing the annual interest rate by 12 and converting it to a decimal. In this case, the monthly interest rate is 7.2% / 12 = 0.006.

Next, we substitute the values into the formula. The loan amount is $256,400, the monthly interest rate is 0.006, and the number of payments is 30 years * 12 months = 360 months.

Plugging in these values into the formula, we have:

Payment = (256,400 * 0.006) / (1 - (1 + 0.006)^(-360))

Calculating this expression, we find that the monthly house payment necessary to amortize the loan is approximately $1,758.70 when rounded to the nearest cent.

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The equation of the plane passing through the points (0, 6, 6), (6, 0, 6), and (6, 6, 0) is: 1296x + 1296y + 1296z - 15552 = 0.

To locate an equation of the aircraft passing thru three given points, we can use the point-normal structure of the equation of a plane.

The ordinary vector of the aircraft can be determined through taking the move product of two vectors in the plane.

Find two vectors in the airplane the usage of the given points:

Vector u = (6, 0, 6) - (0, 6, 6) = (6, -6, 0)

Vector v = (6, 6, 0) - (0, 6, 6) = (6, 0, -6)

Calculate the move product of vectors u and v to locate the ordinary vector of the plane:

Normal vector n = u x v

= (6, -6, 0) x (6, 0, -6)

= (-36, 36, 36)

Step 3: Use one of the given factors (0, 6, 6) and the regular vector (-36, 36, 36) in the point-normal structure of the equation of a plane:

(x - x₁, y - y₁, z - z₁) · (A, B, C) = 0, the place (x₁, y₁, z₁) is the factor and (A, B, C) is the regular vector.

(x - 0, y - 6, z - 6) · (-36, 36, 36) = 0

(-36x, 36y - 216, 36z - 216) · (-36, 36, 36) = 0

Simplifying the equation, we get:

(-36x)(-36) + (36y - 216)(36) + (36z - 216)(36) = 0

1296x + 1296y - 7776 + 1296z - 7776 = 0

1296x + 1296y + 1296z - 15552 = 0

Therefore, an equation of the airplane passing thru the factors (0, 6, 6), (6, 0, 6), and (6, 6, 0) is:

1296x + 1296y + 1296z - 15552 =

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The expected value of the distribution for the random variable x is 10. This means that if we were to repeat the experiment many times, the average value of x would converge to 10, with some fluctuations around this central value due to the randomness of the uniform distribution.

The expected value of a random variable is a measure of the central tendency of its distribution, representing the average value that we would expect to observe over many trials or events. In the case of a uniform distribution, where all values between the minimum and maximum are equally likely, the expected value can be calculated as the average of the two endpoints:
e(x) = (5 + 15) / 2 = 10
Therefore, the expected value of the distribution for the random variable x is 10. This means that if we were to repeat the experiment many times, the average value of x would converge to 10, with some fluctuations around this central value due to the randomness of the uniform distribution.
It is important to note that the expected value is not always equal to the most common or typical value of a distribution, but rather represents a measure of the overall tendency of the data. Other measures such as the variance or standard deviation can provide additional information about the spread or variability of the distribution, which can be useful for making statistical inferences or predictions.

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The area of the rectangle is 200 square centimeters.

The length of the rectangle is twice its width, so the length would be 2W.

The formula for the perimeter of a rectangle is

Perimeter = 2(length + width)

Substituting the values into the formula, we have

60 = 2(2W + W)

Simplifying the equation

60 = 2(3W)

60 = 6W

Dividing both sides of the equation by 6

W = 10

So the width of the rectangle is 10 cm. Since the length is twice the width, the length would be

2 × 10 = 20 cm.

To find the area of the rectangle, we can use the formula

Area = length × width

Substituting the values into the formula, we have

Area = 20 × 10 Area = 200 cm²

Therefore, the area of the rectangle is 200 square centimeters.

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The value of the integral ∫4|x - 2| dx from 0 is equal to 20.

To evaluate the integral ∫4|x - 2| dx from 0, we can interpret it in terms of areas.

The integrand |x - 2| represents the absolute value of the expression (x - 2).

Geometrically, this implies that the function will always be positive and symmetrical about the y-axis.

The interval of integration is from 0, which means we are only concerned with the region to the right of the y-axis.

To evaluate the integral, we need to consider two cases:

When x ≥ 2:

For x ≥ 2, the absolute value of (x - 2) is equal to (x - 2) itself. Therefore, we can rewrite the integral as ∫4(x - 2) dx from 2 to 4.

When x < 2:

For x < 2, the absolute value of (x - 2) is equal to -(x - 2).

Therefore, we can rewrite the integral as ∫4(-(x - 2)) dx from 0 to 2.

Now, let's evaluate each case separately:

Case 1: When x ≥ 2

∫4(x - 2) dx from 2 to 4:

= 4∫(x - 2) dx from 2 to 4

[tex]= 4[(x^2/2 - 2x) |_2^4][/tex]

= 4[(16/2 - 8) - (2/2 - 4)]

= 4[(8 - 8) - (1 - 4)]

= 4(0 - (-3))

= 4(3)

= 12

Case 2: When x < 2

∫4(-(x - 2)) dx from 0 to 2:

= -4∫(x - 2) dx from 0 to 2

[tex]= -4[(x^2/2 - 2x) |_0^2][/tex]

= -4[(4/2 - 4) - (0/2 - 0)]

= -4[(2 - 4) - (0 - 0)]

= -4((-2) - 0)

= -4(-2)

= 8

Now, we can sum up the results of both cases:

12 + 8 = 20.

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In the short-run model for a typical monopolistically competitive firm at profit maximization, the total revenue is $X, total costs are $Y, and the total economic profit is $Z.

In the short-run model for a typical monopolistically competitive firm, profit maximization occurs when marginal revenue (MR) equals marginal cost (MC). The total revenue of the firm, denoted as TR, is the product of the price per unit (P) and the quantity sold (Q). Therefore, TR = P × Q.

To determine the total costs (TC) of the firm, we need to consider both explicit costs (such as wages, rent, and raw material expenses) and implicit costs (such as the opportunity cost of the owner's time and capital invested). TC includes both the variable costs (VC), which change with the level of production, and the fixed costs (FC), which remain constant regardless of the production level. Thus, TC = VC + FC.

Finally, to calculate the total economic profit, we subtract the total costs (TC) from the total revenue (TR). Economic profit (π) is given by the equation π = TR - TC. If the economic profit is positive, the firm is making profits. If it is negative, the firm is incurring losses. If it is zero, the firm is just breaking even.

Overall, in the short-run model for a monopolistically competitive firm at profit maximization, the total revenue (TR) is $X, the total costs (TC) are $Y, and the total economic profit (π) is $Z. These figures are crucial for understanding the financial performance of the firm and its position in the market.

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[tex]x = \frac{15}{10} \times y[/tex]

.

We need to show that all of the following conditions hold: a) The outcomes for x are countable and represent specific points on the real number line. b) The probability values, P(x), are between 0 and 1 for all x. c) The sum of all the probabilities, ΣP(X), is equal to 1. The correct answer is option d: All of the above.

In order for a probability distribution to be valid, it must satisfy certain criteria. First, the outcomes for x must be countable, meaning that they are specific points on the real number line. This ensures that the distribution is well-defined and that each outcome has a corresponding probability assigned to it.

Second, the probability values, P(x), must be between 0 and 1 for all x. Probabilities represent the likelihood of each outcome occurring, and they cannot be negative or greater than 1. This condition ensures that the probabilities are within a valid range.

Lastly, the sum of all the probabilities in the distribution must be        equal to 1. This means that when you add up the probabilities of all possible outcomes, the total probability should equal 1, indicating that one of the outcomes will occur.

By verifying that all of these conditions hold for the given probability distribution, we can conclude that it is a valid discrete probability distribution.

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In a 6×8 matrix the col A is not equal to ℝ⁵ but is at most ℝ⁵ and the nul A is equal to ℝ³.

What is a matrix ?

A matrix is a rectangular array of numbers or elements arranged in rows and columns.

In order to determine whether the column space of matrix A (col A) is ℝ⁵ and the null space of matrix A (nul A) is ℝ³, we need to examine the dimensions of the matrix and the number of pivot columns.

Given that matrix A is a 6×8 matrix and has five pivot columns, it means that there are five linearly independent columns in the matrix A. The pivot columns are the columns that contain the leading 1's in the row echelon form of the matrix.

1. col A = ℝ⁵:

Since there are five pivot columns in matrix A, the column space of matrix A (col A) is the span of these five columns. The column space is the subspace of ℝ⁶ spanned by the linearly independent columns. Since there are only five linearly independent columns, the column space of matrix A cannot be equal to ℝ⁵. It can be at most ℝ⁵, but not necessarily equal to it.

2. nul A = ℝ³:

The null space of matrix A (nul A) consists of the solutions to the homogeneous equation A*x = 0, where x is a vector. The null space is the subspace of ℝ⁸ spanned by the solutions to this equation. The dimension of the null space is equal to the number of free variables in the solution.

Since matrix A has eight columns, the null space of matrix A can have at most eight dimensions (ℝ⁸). However, the fact that there are five pivot columns means that there are three free variables in the solution to A*x = 0. Therefore, the dimension of the null space is equal to three, which means nul A = ℝ³.

In summary:

- col A is not equal to ℝ⁵ but is at most ℝ⁵.

- nul A is equal to ℝ³.

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Answer:

x≈4.63902101

Step-by-step explanation:

True.

The method of least squares is used in regression analysis to determine the best-fitting line or curve that minimizes the sum of squared errors (SSE). The SSE is calculated by summing the squared differences between the observed values and the corresponding predicted values.

The goal of the least squares method is to find the line that minimizes this sum, indicating the line that has the smallest overall error.

Additionally, the least squares method ensures that the regression line has an average error of zero by taking into account the overall deviation from the line. This means that, on average, the predicted values from the regression line will be equal to the observed values.

So, it is true that the method of least squares specifies that the regression line has an average error of 0 and an SSE that is minimized.

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If the taxi operated every day, the annual profit per taxicab license would be $8,030. The market price of a taxicab license, assuming a 10% interest rate and constant conditions, would be $220. The number of additional licenses needed to reduce the taxicab price per ride to $5 would depend on the demand and supply conditions.

(1) The competitive equilibrium price per ride is determined by setting the price equal to the marginal cost. Thus, the competitive equilibrium price is $5. The equilibrium number of rides per day can be calculated by substituting the price into the demand function D(P) = 1200 - 20p, giving 1,000 rides.

(2) In 1995, the demand curve changed to D(n) = 1220 - 20p. By setting the demand equal to the supply, the equilibrium price of a ride in 1995 can be determined, which is $6.10.

(3) The profit per ride in 1995, can be calculated by subtracting the marginal cost ($5) from the equilibrium price ($6.10), resulting in a profit of $1.10 per ride. The profit per taxicab license per day is obtained by multiplying the profit per ride by the taxi capacity, resulting in $22. If the taxi operates every day, the annual profit per taxicab license will be $8,030.

(4) The market price of a taxicab license can be determined by discounting the annual profit per taxicab license, resulting in $220.

(5) The exact number of additional licenses needed would depend on the specific demand and supply conditions.

(6) The value of a taxicab license at the new fare of $5 would depend on the expected profits generated at that price.

(7) It depends on the perceived impact on their profits and on their individual valuations and expectations.

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The equation of the parabola in standard form is f(x) = -(x - 3)² + 25

Here, we have,

the vertex of the parabola is given as (3, 25), we know that the equation of the parabola can be written in vertex form as:

f(x) = a(x - 3)² + 25

where a is a constant that determines the shape of the parabola.

The point (6, 16) lies on the parabola

so we can substitute x = 6 and y = 16 into the equation above to find 'a'.

16 = a(6 - 3)² + 25

-9 = 9a

Dividing both sides by 9, we get:

a = -1

Now we know that the equation of the parabola is:

f(x) = -(x - 3)² + 25

Hence, the equation of the parabola in standard form is

f(x) = -(x - 3)² + 25

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False.

The statement is not necessarily true. The number of levels of observed x-values does not have to be equal to the order of the polynomial in x that you want to fit.

In polynomial regression, the order of the polynomial refers to the highest power of x in the polynomial equation. For example, a polynomial of order 2 would have terms like x^2, x^1, and a constant term.

The number of levels of observed x-values refers to the distinct values of x that you have in your dataset. It represents the range or diversity of x-values observed.

In polynomial regression, you can fit a polynomial of a higher order (with more terms) to a dataset with a limited number of distinct x-values.

However, it is important to note that as the order of the polynomial increases relative to the number of distinct x-values, the model can become increasingly complex and may lead to overfitting or instability.

Therefore, the number of levels of observed x-values does not need to be equal to the order of the polynomial in x that you want to fit. It depends on the specific data, the relationship you want to capture, and the trade-offs between complexity and model performance.

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The equation that can be used is the exponential growth equation p = p0 * e^(kt), where p0 is the initial population, e is Euler's number, k is the growth rate constant, and t is the number of years after.

In this case, we need to solve for t when p = 3p0. The given formula for modeling population growth, p = p0 * e^(kt), is an exponential growth equation. In this equation, p0 represents the initial population, e is Euler's number (approximately 2.71828), k is the growth rate constant, and t is the number of years after.

To find the number of years after which the population triples, we need to solve for t when p = 3p0. Substituting 3p0 for p in the equation, we get 3p0 = p0 * e^(kt). By canceling out p0 on both sides, we have 3 = e^(kt). To solve for t, we take the natural logarithm of both sides to eliminate the exponential term. Applying the natural logarithm to both sides gives us ln(3) = ln(e^(kt)). Since the natural logarithm and exponential functions are inverse operations, the exponential term simplifies to kt. Therefore, t = ln(3) / k.

By substituting the specific growth rate constant value, we can determine the exact number of years after which the population will triple.

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For the points A(-4,10) and B(6), the calculations are as follows: Slope = 2, Midpoint = (1,5), Distance = 10√2. For the points F(-1,3) and G(9,-3), the calculations are as follows: Slope = -1, Midpoint = (4,0), Distance = 10√2. Regarding the points R(-5,8) and T(3,14), the midpoint of RT is (-1,11).

1. For the points A(-4,10) and B(6):

  - Slope: The formula for slope is (y₂ - y₁) / (x₂ - x₁). Plugging in the values, we get (10 - 6) / (6 - (-4)) = 4 / 10 = 2.

  - Midpoint: The midpoint formula is ((x₁ + x₂) / 2, (y₁ + y₂) / 2). Substituting the values, we have ((-4 + 6) / 2, (10 + 6) / 2) = (1, 5).

  - Distance: The distance formula is √[(x₂ - x₁)² + (y₂ - y₁)²]. Plugging in the values, we get [tex]\sqrt{(6 - (-4)^{2} + (10 - 10)^{2} } =10\sqrt{2}[/tex].

2. For the points F(-1,3) and G(9,-3):

  - Slope: Using the slope formula, we get (-3 - 3) / (9 - (-1)) = -6 / 10 = -1.

  - Midpoint: Applying the midpoint formula, we get ((-1 + 9) / 2, (3 + (-3)) / 2) = (4, 0).

  - Distance: Using the distance formula, we get √[(9 - (-1))² + (-3 - 3)²] = √(10²) = 10√2.

3. For the points R(-5,8) and T(3,14):

  - Midpoint: The midpoint formula gives us ((-5 + 3) / 2, (8 + 14) / 2) = (-1, 11).

Thus, the midpoint of RT is (-1, 11).

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Answer: I would expect them to win 80 games!

Step-by-step explanation:

1) Multiply 0.4 by 200.

       200 · 0.4 = 80

True. The loan-to-value ratio is a calculation that compares the amount of the loan being taken out for a real estate investment project to the value of the property being purchased. This ratio is often used to determine the amount of leverage being used in the investment, with higher ratios indicating more leverage and potentially higher risk.

The LTV ratio indicates the percentage of the property's value that is financed through debt. Higher LTV ratios indicate a higher level of leverage, meaning that a larger portion of the property's value is funded through borrowed money. LTV ratios are commonly used by lenders to assess the risk associated with a real estate investment and determine the terms of the loan.

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To find the zeros of a polynomial, set the polynomial equal to zero and solve for the variable.

To find the zeros of a polynomial, follow these step-by-step instructions:

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The exact value of sec(13π/4) is √2.

To evaluate sec(13π/4) using reference angles, we first need to determine the reference angle for the given angle.

The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis.

In this case, 13π/4 is in the third quadrant of the unit circle, where the x-coordinate is negative and the y-coordinate is negative.

The reference angle for an angle in the third quadrant is the angle formed between the terminal side and the nearest x-axis line in the first quadrant.

The nearest x-axis line in the first quadrant is π/4 (45 degrees).

Since secant is the reciprocal of cosine, we can use the cosine of the reference angle to find the secant value.

The cosine of π/4 is equal to √2/2.

Reciprocal of √2/2 is 2/√2, which can be simplified as √2.

Therefore, sec(13π/4) = √2.

Hence, the exact value of sec(13π/4) is √2.

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Answer:

12.9 cm

Step-by-step explanation:

area of full circle = 2 X 65.8 = 131.6cm².

131.6 = π r² = (3.14r²)

r² = 131.6/3.14 = 41.9108....

r = 6.47.

diameter = 2r = 12.94 = 12.9 cm to nearest tenth

A cost that varies depending on the values of decision variables is a variable cost. This type of cost changes in proportion to changes in the production or sales volume.

Variable costs are expenses that fluctuate with changes in the level of production or sales. These costs are directly tied to the quantity of goods or services produced or sold, and they increase or decrease as the production or sales volume changes. Common examples of variable costs include materials, labor, and direct expenses associated with producing a product or providing a service. As the production or sales volume increases, the variable cost per unit decreases, due to economies of scale. Conversely, as production or sales volume decreases, the variable cost per unit increases, due to diseconomies of scale. Understanding variable costs is essential for businesses to accurately calculate their costs of goods sold, determine their break-even point, and make informed decisions about pricing and production levels.

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The equation for the plane tangent to each surface - -10x + 6y + z - 5 = 0

What is Plane Tangent?

a tangent is a line that touches a curve at exactly one point without intersecting it. Similarly, a plane tangent is a plane that touches a surface at only one point without intersecting it further. This concept is commonly used in calculus and differential geometry to study the behavior of curves and surfaces.

We usually use partial derivatives of the equation of the surface to find the equation of the plane tangent to the surface at a point. The normal vector to the surface at that point is obtained by taking the cross product of the partial derivatives evaluated at that point. Then, using the coordinates of the point and the normal vector, we can determine the equation of the plane using the point-normal form or the general form of the plane equation.

It is important to note that the tangent plane equation depends on the specific point on the surface where the tangent plane is desired. Different points on the surface will have different tangent planes associated with them.

To find the equation for the plane tangent to the surface z = f(x, y) at a point (a, b, c), we need to use the partial derivatives of f with respect to x and y at the point (a, b) to find the normal vector to the plane.

Then we can use the point-normal form of the equation of a plane to write the equation.

First, we need to find the partial derivatives of f(x, y) = x^2 + y^3 - 6xy with respect to x and y:

fx = 2x - 6y

fy = 3y^2 - 6x

Then we can evaluate these partial derivatives at the point (1, 2) to get the normal vector:

n = <fx(1, 2), fy(1, 2)> = <2(1) - 6(2), 3(2)^2 - 6(1)> = <-10, 6>

So the equation of the plane tangent to the surface z = f(x, y) = x^2 + y^3 - 6xy at the point (1, 2, 3) is:

-10(x - 1) + 6(y - 2) + (z - 3) = 0

Simplifying, we get:

-10x + 6y + z - 5 = 0

So the equation of the plane tangent to the surface z = x^2 + y^3 - 6xy at the point (1, 2, 3) is -10x + 6y + z - 5 = 0

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Answer:

0.7 is the probability they would allow it

Step-by-step explanation:

0.3 is the probability out of 1

so 1-0.3 is the probability they would allow it

1-0.3=0.7