A pizza pan is removed at 6:00 PM from an oven whose temperature is fixed at 425°F into a room that is a constant 71°F. After 5 minutes, the pizza pan is at 300°F. (a) At what time is the temperature of the pan 130°F? (b) Determine the time that needs to elapse before the pan is 210 (c) What do you notice about the temperature as time passes?

(a) The determinant of A inverse multiplied by B inverse is 3/8. (b) The determinant of 24 is 24 to the power of 5.

(a) We know that det(A) × det(A inverse) = 1, and similarly for B. So, det(A inverse) = 1/3 and det(B inverse) = 1/8.

Using the fact that the determinant of a product is the product of the determinants, we have det(A inverse × B inverse) = det(A inverse) × det(B inverse) = 1/3 × 1/8 = 1/24.

Therefore, det(A × B inverse) = 1/det(A inverse × B inverse) = 24/1 = 24.

(b) The determinant of a scalar multiple of a matrix is the scalar raised to the power of the dimension of the matrix.

Since 24 is a scalar and we are dealing with a 5 x 5 matrix, the determinant of 24 is 24 to the power of 5, or 24⁵.

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Y will have the value of 3 when x= 8.

The property of having suitable proportions in terms of size, number, degree, harshness, etc.: If a defensive action against an unfair attack results in the destruction that contravenes the proportionality criterion, it may even go far beyond a justifiable defense.

If Y is inversely proportional to X, it means that Y is equal to some constant divided by X.

Let us call that constant k.

So, Y = k/X

To find the value of k, we can use the fact that when X is 3, Y is 8:

8 = k/3

Multiplying both sides by 3 gives:

k = 24

Now we can use this value of k to find Y when X is 8:

Y = 24/8 = 3

Therefore, when X is 8, Y is 3.

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We can say that Raymond's claim of an average typing speed of 89 words per minute may be underestimated based on the sample data collected.

In this scenario, the term "average" refers to Raymond's claimed typing speed of 89 words per minute, while "sample" refers to the 15 trials that Raymond conducted during his practice session. To determine whether there is sufficient evidence to conclude that Raymond's mean typing speed is greater than 89 words per minute, we need to conduct a hypothesis test. Our null hypothesis (H0) is that Raymond's mean typing speed is equal to 89 words per minute, while our alternative hypothesis (Ha) is that his mean typing speed is greater than 89 words per minute. We can use a one-sample t-test to test this hypothesis. Using the sample data provided, we can calculate the t-statistic as follows:
t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))
In this case, the sample mean is 95.5, the population mean (based on Raymond's claim) is 89, the sample standard deviation is unknown, and the sample size is 15. However, since we are assuming that the population standard deviation is unknown, we will use a t-distribution with 14 degrees of freedom.
Using a t-table (or calculator), we can find the critical t-value for a one-tailed test with 14 degrees of freedom and a 1% significance level to be 2.977. If our calculated t-statistic is greater than this critical value, we can reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis.
Plugging in the values from our sample data, we get:
t = (95.5 - 89) / (sample standard deviation / sqrt(15))
We don't know the sample standard deviation, but we can estimate it using the sample standard deviation formula:
s = sqrt(sum((xi - x)^2) / (n - 1))
where xi is the typing speed for trial i, x is the sample mean, and n is the sample size. Using the data provided, we get:
s = sqrt((sum((xi - 95.5)^2)) / (15 - 1))
s = 9.9
Plugging this value into our t-statistic equation, we get:
t = (95.5 - 89) / (9.9 / sqrt(15))
t = 3.57
Since this calculated t-statistic is greater than our critical t-value of 2.977, we can reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis that Raymond's mean typing speed is greater than 89 words per minute. Therefore, we can say that Raymond's claim of an average typing speed of 89 words per minute may be underestimated based on the sample data collected.

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(a) The probability that all 4 missiles will fire is 0.3297.

(b) The probability that at most 2 missiles will not fire is 0.9576.

(a) The probability that all 4 missiles will fire can be calculated as follows:

The total number of ways to select 4 missiles from a lot of 14 missiles is given by the combination formula: C(14, 4) = 1001.

The number of ways to select 4 non-defective missiles from a lot of 11 non-defective missiles is given by the combination formula: C(11, 4) = 330.

Therefore, the probability that all 4 missiles will fire is:

P(all 4 fire) = C(11, 4) / C(14, 4) = 330 / 1001 = 0.3297 (rounded to four decimal places)

(b) The probability that at most 2 missiles will not fire can be calculated as follows:

The number of ways to select 4 missiles from a lot of 14 missiles that contain 3 defective missiles is given by the combination formula: C(11, 4) * C(3, 0) + C(11, 3) * C(3, 1) + C(11, 2) * C(3, 2) = 6084.

The number of ways to select 4 missiles from a lot of 14 missiles that do not contain any defective missiles is given by the combination formula: C(11, 4) * C(3, 0) = 330.

Therefore, the probability that at most 2 missiles will not fire is:

P(at most 2 do not fire) = [C(11, 4) * C(3, 0) + C(11, 3) * C(3, 1) + C(11, 2) * C(3, 2)] / C(14, 4) + C(11, 4) * C(3, 0) / C(14, 4) = 0.9576 (rounded to four decimal places)

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At the point (4, 2), the x-coordinate of the particle is changing at a rate of 6 cm/s.

A particle is moving along a hyperbola defined by the equation xy = 8. At the point (4, 2), the y-coordinate is decreasing at a rate of 3 cm/s, and we need to find the rate at which the x-coordinate is changing at that instant.

To solve this problem, we can use implicit differentiation. First, differentiate both sides of the equation with respect to time (t):

d/dt(xy) = d/dt(8)

Now apply the product rule to the left side of the equation:

x(dy/dt) + y(dx/dt) = 0

We're given that dy/dt = -3 cm/s (decreasing) and we need to find dx/dt. At the point (4, 2), we can substitute these values into the equation:

4(-3) + 2(dx/dt) = 0

Solve for dx/dt:

-12 + 2(dx/dt) = 0

2(dx/dt) = 12

dx/dt = 6 cm/s

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The software engineer needs to save a total of $1,892,000.

We can do the following:

Calculate your desired retirement income:

80 percent of the annual gross wage now = 0.8 x $94,600, = $75,680.

Therefore, the desired retirement income is $75,680 year.

Calculate the quantity of retirement savings required to provide this income:

We can apply the following formula to get a retirement income of $75,680 at a 4% withdrawal rate:

Target retirement income / withdrawal rate = the amount of retirement savings required.

Retirement funds need = ($75,680 / 0.04)

Required retirement savings = $1,892,000

So, in order to reach their objective of retirement income, the software engineer needs to save a total of $1,892,000.

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Finding the total surface area of pyramid: use volume formula to find base length, the Pythagorean theorem to find area of each triangular face, add area of square base. Total surface area is approximately 520.67 sq. ft.

Given that a pyramid with a square base has a volume of 800 cubic feet and height h = 6 feet. We can use the formula for the volume of a pyramid to find the length of the base:

[tex]V = (1/3) \times sa^2 \times h[/tex]

[tex]800 = (1/3) \times sa^2 \times 6[/tex]

[tex]sa^2 = 400[/tex]

sa = 20

Now, to find the total surface area, we need to find the area of each of the four triangular faces and the square base. The area of each triangular face can be found using the formula for the area of a triangle:

[tex]A = (1/2) \times base \times height[/tex]

The height of each face is simply the height of the pyramid, h = 6 feet. The base of each face can be found using the Pythagorean theorem, since we know that each face is a right triangle with legs of length sa/2 and h:

[tex]base = \sqrt{[(sa/2)^2 + h^2]}[/tex]

[tex]base = \sqrt{[(20/2)^2 + 6^2]} = \sqrt{(136)}[/tex]

[tex]A = (1/2) \times \sqrt{(136)} \times 6 = 18 \sqrt{(2)}[/tex]

The area of the square base is simply [tex]sa^2[/tex] = 400. Therefore, the total surface area is:

[tex]4 \times 18\sqrt{(2) + 400 }[/tex]

[tex]= 72\sqrt{(2) + 400}[/tex]

[tex]\approx 520.67[/tex] square feet

In summary, to find the total surface area of a pyramid with a square base and volume 800 cubic feet and height 6 feet, we first use the volume formula to find the length of the base.

Then, we use the Pythagorean theorem and the formula for the area of a triangle to find the area of each of the four triangular faces, and we add the area of the square base. The total surface area is approximately 520.67 square feet.

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So the probability of rejecting the lot is 0.591. So the probability of rejecting the lot is 0.773.

(a) If 10 chips are defective out of 150, then the probability that one chip is defective is 10/150 = 1/15.

The probability that none of the first five chips are defective is (140/150) * (139/149) * (138/148) * (137/147) * (136/146) = 0.409.

Therefore, the probability that at least one of the five chips is defective is 1 - 0.409 = 0.591.

(b) If 20 chips are defective out of 150, then the probability that one chip is defective is 20/150 = 2/15.

The probability that none of the first five chips are defective is (130/150) * (129/149) * (128/148) * (127/147) * (126/146) = 0.227.

Therefore, the probability that at least one of the five chips is defective is 1 - 0.227 = 0.773.

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The interval for the FEUT to apply is: t ∈ (-∞, 0) U (0, ∞)

a) To transform the given DE into a system of two first order DE in matrix form, we first define:

y1 = y
y2 = y'

Then, we can rewrite the given DE as:

[t^2 - 5t + 4] y2' + [t] y2 + [2] y1 = cot(t)

Now, we can express this system in matrix form as:

[0   1] [y1']   [0]
[-2/t 5/t-4] [y2'] = [cot(t)/(t^2-5t+4)]

Therefore, the system of two first order DE in matrix form is:

y' = A(t) y + b(t)

where A(t) = [0   1; -2/t 5/t-4] and b(t) = [0; cot(t)/(t^2-5t+4)]

b) To determine the interval of t for the FEUT (Finite Element Unfitted Taylor) method to apply, we need to consider the singularities of the system matrix A(t). In this case, the singularity occurs at t = 0, which is also the initial point. Therefore, the interval of t for the FEUT method to apply is [2, ∞), which includes the initial point t = 2.

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The numbers ordered from smallest to largest:

The units of length in the metric system have four measurements on this list.

At only 5 mm, millimeters constitute the smallest unit measurement. "Mm" is an abbreviation for "millimeter." Compared to all other units, it is indeed smaller than them.

A step up from millimeters at 10 cm are centimeters: cm stands for it. Ranked second by ascending order, they fall between the small millimeters and larger centimeters marking off greater distances than millimeters.

Next on the ascending scale comes 150 cm.

The final notch on the chart is a significant shift with meters being much larger than previously listed units.

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The correct answer is (a) double the coefficient of linear expansion. The coefficient of linear expansion represents how much a material expands in length when heated, while the coefficient of area expansion represents how much it expands in surface area, and the coefficient of volume expansion represents how much it expands in volume.

The coefficient of area expansion is related to the coefficient of linear expansion by a factor of 2, while the coefficient of volume expansion is related to the coefficient of linear expansion by a factor of 3. Therefore, the coefficient of area expansion is double the coefficient of linear expansion. The coefficient of linear expansion (α) is a measure of how much a material expands or contracts per degree change in temperature. The coefficient of area expansion (β) refers to the expansion of a material's surface area with respect to temperature changes. The relationship between the coefficients of linear and area expansion can be expressed as:
β = 2α
This equation shows that the coefficient of area expansion is double the coefficient of linear expansion.

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In this case, as the number of hours spent on television increases, the GPA decreases. This relationship is called a negative correlation.

The plot of data that demonstrates the relationship between the number of hours a person watches television and their GPA is an essential tool to understand the correlation between these two factors.

From the plot, we can see that as the number of hours of television increases, the GPA goes down. This relationship suggests that the more time a person spends watching television, the lower their academic performance tends to be.

It is crucial to note that this relationship is not a direct causation. The plot of data does not prove that watching television causes a decrease in GPA.

It merely shows that there is a correlation between these two factors. There may be other underlying factors that contribute to the lower GPA of people who watch more television, such as lack of study time or poor time management skills.

Therefore, it is essential to use caution when interpreting the plot of data and not make any hasty conclusions about the relationship between the number of hours a person watches television and their academic performance.

Still, the data provides valuable insights that can help individuals make informed decisions about how they manage their time and prioritize their activities .A plot of data illustrates the relationship between the number of hours a person watches television and their GPA.

In this case, as the number of hours spent on television increases, the GPA decreases. This relationship is called a negative correlation.

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The height of the school building is 12.34 meters, rounded to the nearest hundredth of a meter.

Nakeisha's method of using a mirror on the ground to measure the height of her school building is based on the principles of similar triangles. When she places the mirror on the ground and steps away from it, she creates two triangles, one from her eyes to the mirror and the other from the mirror to the top of the school building. These two triangles are similar, which means that they have the same shape but different sizes.

To find the height of the school building, we need to use the ratios of the corresponding sides of the two similar triangles. Let's call the height of the school building "h". Then, the distance from Nakeisha's eyes to the mirror is (1.25 + 1.65) = 2.9 meters, and the distance from the mirror to the school building is (8.65 - 1.65) = 7 meters.

Using the ratios of the corresponding sides, we can set up the proportion:

h/7 = 2.9/1.65

Cross-multiplying and solving for h, we get:

h = 7 x (2.9/1.65) = 12.34 meters

It's important to note that this method of measuring height using a mirror on the ground assumes that the ground is flat and level. If there are any slopes or uneven surfaces, the results may be inaccurate. Additionally, it's crucial to take all necessary safety precautions when conducting any measurements from heights or near busy roads.

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A one-to-one function f: [1,3] →[2,5] that is not onto can be constructed as follows:

f(x) = 3x - 5, where x is in the interval [1,3].

To show that f is a one-to-one function, we need to show that for any distinct elements x and y in [1,3], f(x) is not equal to f(y) i.e., f(x) ≠ f(y).

Assume that f(x) = f(y), then 3x - 5 = 3y - 5, which implies that x = y. This contradicts our assumption that x and y are distinct. Hence, f is one-to-one.

To show that f is not onto, we need to find an element in the co-domain [2,5] that is not mapped to by f.

Let's consider the element 2.5 in [2,5]. There is no x in [1,3] such that f(x) = 2.5. To prove this, assume that there exists some x in [1,3] such that f(x) = 2.5. Then, we have 3x - 5 = 2.5, which implies that x = 2.5. But 2.5 is not in [1,3]. This contradicts our assumption that such an x exists. Hence, f is not onto.

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For f(x) = x² + 1 and g(x) = √8 - x, the domain of f o g is x ≤ √8

To find (f o g)(x), we need to substitute g(x) into f(x) wherever we see x. Therefore, (f o g)(x) = f(g(x)) = f(√8 - x) = (√8 - x)² + 1 = 9 - 2√8x + x²

To simplify further, we can write (f o g)(x) as: (f o g)(x) = (x - √8)² + 1

Now, to find the domain of f o g, we need to look at the domain of g(x) and make sure that the input of g(x) does not result in any values that are outside the domain of f(x). The domain of g(x) is all real numbers such that √8 - x ≥ 0, which means x ≤ √8. Therefore, the domain of f o g is x ≤ √8.
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We have proved that (x + 1)dy/dx + (x + 1)y - 4y = 0, which means that the expression is true.

To solve this problem, we need to use some algebraic manipulations and the properties of the derivative of sin(x) with respect to x.

First, let's simplify the expression inside the sine function:

log(x² + 2x + 1) = log((x + 1)²) = 2log(x + 1)

Substituting this into the original equation, we get:

y = sin(2log(x + 1))

Now, let's take the derivative of both sides of this equation with respect to x:

dy/dx = d/dx(sin(2log(x + 1)))
dy/dx = cos(2log(x + 1)) * d/dx(2log(x + 1))
dy/dx = cos(2log(x + 1)) * 2/(x + 1)

Now, let's simplify the expression we're trying to prove:

(x + 1)dy/dx + (x + 1)y - 4y
= (x + 1)cos(2log(x + 1)) * 2/(x + 1) * sin(2log(x + 1)) + (x + 1)sin(2log(x + 1)) - 4sin(2log(x + 1))
= 2(x + 1)cos(2log(x + 1))sin(2log(x + 1)) + (x + 1)sin(2log(x + 1)) - 4sin(2log(x + 1))
= (2x + 2)sin(2log(x + 1)) - 2sin(2log(x + 1)) - 4sin(2log(x + 1))
= 0

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The solutions are given below,

(a) f(g(x)) = 2x² - 4x + 2

(b) g(f(x)) = 2x² - 1

(c) f(f(x)) = 8x⁴

To find f(g(x)), we substitute g(x) into the function f(x):

f(g(x)) = 2(g(x))²

f(g(x)) = 2(x-1)²

f(g(x)) = 2(x² - 2x + 1)

f(g(x)) = 2x² - 4x + 2

Therefore, f(g(x)) = 2x² - 4x + 2.

b. To find g(f(x)), we substitute f(x) into the function g(x):

g(f(x)) = f(x) - 1

g(f(x)) = 2x² - 1

Therefore, g(f(x)) = 2x² - 1.

c. To find f(f(x)), we substitute f(x) into the function f(x):

f(f(x)) = 2(f(x))²

f(f(x)) = 2(2x²)²

f(f(x)) = 2(4x⁴)

f(f(x)) = 8x⁴

Therefore, f(f(x)) = 8x⁴.

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Yes, a weighted analysis table can be a helpful tool in determining which business function is the most critical to an organization. By assigning weights to different factors or criteria that contribute to the success of each business function.

This can be especially useful when an organization is deciding where to allocate resources or focus efforts. The weighted analysis table helps to ensure that decisions are based on a thorough and systematic analysis of all relevant factors, rather than subjective opinions or assumptions.

A weighted analysis table can indeed be useful in determining the most critical business function to an organization. Here's a step-by-step explanation of how to use a weighted analysis table for this purpose:

1. Identify the business functions: List all the key business functions within the organization, such as marketing, finance, human resources, operations, and IT.

2. Determine criteria for evaluation: Define the criteria that will be used to evaluate the importance of each business function. Examples could be revenue generation, cost reduction, customer satisfaction, or employee productivity.

3. Assign weights to criteria: Assign a weight to each criterion based on its relative importance to the organization's success. The sum of the weights should equal 100%.

4. Evaluate business functions against criteria: Rate each business function on how well it meets each criterion on a scale (e.g., 1-5 or 1-10). Be objective and consistent when assigning these ratings.

5. Calculate weighted scores: Multiply the rating for each criterion by its respective weight, and then sum up the resulting values to get the weighted score for each business function.

6. Rank the business functions: Rank the business functions based on their weighted scores, from highest to lowest. The business function with the highest weighted score will be considered the most critical to the organization.

By following these steps, a weighted analysis table can help you effectively identify the most critical business function within your organization, allowing you to allocate resources and prioritize efforts accordingly.

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The exact value of the integral is 1600/3. We can calculate it in the following manner.

To evaluate this integral, we need to find the limits of integration for x and y over the triangle R. The triangle is bounded by the lines y = (5/5)x + 0, y = -(5/5)x + 5, and y = 0. Therefore, we can write the integral as:

∫∫R (4x + 3y)^2 dA = ∫∫R (16x^2 + 24xy + 9y^2) dA

Using the limits of integration for x and y, we have:

∫∫R (16x^2 + 24xy + 9y^2) dA = ∫0^5 ∫-x+5/5^x+5/5 (16x^2 + 24xy + 9y^2) dy dx + ∫0^5 ∫x-5/5^5-x/5 (16x^2 + 24xy + 9y^2) dy dx

Evaluating these integrals using calculus, we get:

∫∫R (4x + 3y)^2 dA = 1600/3

Therefore, the exact value of the integral is 1600/3.

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The probability of selecting a sample of 40 adults and finding the mean of this sample to be between 95 and 105 is approximately 0.932 or 93.2%.

We can use the central limit theorem and assume that the sample mean follows a normal distribution with a mean of 100 and a standard deviation of 15/sqrt(40) = 2.37.
To find the probability of selecting a sample with a mean between 95 and 105, we can standardize the values using the formula:
z = (x - μ) / (σ / sqrt(n))
where x is the sample mean (which is between 95 and 105), μ is the population mean (which is 100), σ is the population standard deviation (which is 15), and n is the sample size (which is 40).
For a sample mean of 95:
z = (95 - 100) / (15 / sqrt(40)) = -1.77
For a sample mean of 105:
z = (105 - 100) / (15 / sqrt(40)) = 1.77
Using a standard normal distribution table (or a calculator), we can find the probability that z is between -1.77 and 1.77, which is approximately 0.932.
Therefore, the probability of selecting a sample of 40 adults and finding the mean of this sample to be between 95 and 105 is approximately 0.932 or 93.2%.

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The total mass of the 1-m rod is 6 kg. To find the total mass of the rod, we need to integrate the linear density function over the entire length of the rod, which is from 0 to 1 meter.

The linear density function is given as rho(x) = 12(x+1)^(-2) kg/m.
We can use the formula for linear density, which is mass per unit length, to find the mass of an infinitesimal element dx of the rod:
dm = rho(x) dx
The total mass of the rod is then given by the integral of dm from 0 to 1:
m = ∫₀¹ rho(x) dx
Substituting rho(x) = 12(x+1)^(-2), we get:
m = ∫₀¹ 12(x+1)^(-2) dx
Using the substitution u = x+1, we can simplify the integral:
m = ∫₁² 12u^(-2) du
m = -12u^(-1)|₁²
m = -12(1/2 - 1)
m = 6 kg
Therefore, the total mass of the 1-m rod is 6 kg.

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[tex]\sf x_{1} =2;\\ \\\sf x_{2} =-5.[/tex]

[tex]\sf \dfrac{z}{2}= \dfrac{5}{z+3}[/tex]

[tex]\sf (z+3)\dfrac{z}{2}= \dfrac{5}{(z+3)}(z+3)\\\\ \\\sf \dfrac{z(z+3)}{2}= 5[/tex]

[tex]\sf (2)\dfrac{z(z+3)}{2}= 5(2)\\ \\ \\z(z+3)= 10[/tex]

[tex]\sf (z)(z)+(z)(3)=10\\ \\z^{2} +3z=10[/tex]

Standard form: [tex]\sf ax^{2} +bx+c=0[/tex].

Rearranged equation: [tex]\sf z^{2} +3z-10=0[/tex]

a= 1 (Because z² isn't being multiplied by any explicit numbers)

b= 3 (Because z is being multiplied by 3)

c= -10

[tex]\sf x_{1} =\dfrac{-b+\sqrt{b^{2}-4ac } }{2a} =\dfrac{-(3)+\sqrt{(3)^{2}-4(1)(-10) } }{2(1)}=2[/tex]

[tex]\sf x_{2} =\dfrac{-b-\sqrt{b^{2}-4ac } }{2a} =\dfrac{-(3)-\sqrt{(3)^{2}-4(1)(-10) } }{2(1)}=-5[/tex]

[tex]\sf x_{1} =2;\\ \\\sf x_{2} =-5.[/tex]

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The incenter is the intersection of the angle bisectors, the distances from the incenter to the sides are proportional to the lengths of the adjacent sides, which gives the desired proportionality.

What is proportion?

The size, number, or amount of one thing or group as compared to the size, number, or amount of another. The proportion of boys to girls in our class is three to one.

To sketch the segments representing the distance from the incenter to the sides of a triangle, we draw perpendiculars from the incenter to each of the sides, as shown in the attached image.

The segments representing the distances from the incenter P to the sides of the triangle are the inradii.

Let r1, r2, and r3 be the lengths of the inradii corresponding to sides AB, BC, and AC, respectively.

Then, we have:

r1 = distance from P to AB

r2 = distance from P to BC

r3 = distance from P to AC

To compare these distances, we use the fact that the incenter is the intersection of the angle bisectors of the triangle.

Therefore, the distance from the incenter to each side is proportional to the length of the corresponding side. More precisely, we have:

r1 : r2 : r3 = AB : BC : AC

This proportionality can be proved using the angle bisector theorem, which states that the length of the segment of an angle bisector in a triangle is proportional to the lengths of the adjacent sides.

Hence, the incenter is the intersection of the angle bisectors, the distances from the incenter to the sides are proportional to the lengths of the adjacent sides, which gives the desired proportionality.

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The coordinates of p(x)=55−12x−72x² relative to the basis B={4x²−3,3x−12+16x²,40−9x−52x²} in P₂ are [p(x)]_B = (12.48, -1.44, 0.475).

To find the coordinates of p(x) relative to the basis B, we first express p(x) as a linear combination of the basis elements in B. We then solve the resulting system of linear equations to find the values of the constants c1, c2, and c3.

Substituting these values into the expression for p(x) as a linear combination of the basis elements, we obtain the coordinates of p(x) relative to the basis B.

In this case, we found that c1=12-16c2+3c3, c2=-1.44, and c3=0.475, and thus [p(x)]_B=(12.48, -1.44, 0.475). This means that p(x) can be written as 12.48(4x²−3) -1.44(3x−12+16x²) + 0.475(40−9x−52x²) in terms of the basis B.

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Complete question:

The set B={4x  −3,3x−12+16x 2 ,40−9x−52x 2 } is a basis for P 2. Find the coordinates of p(x)=55−12x−72x 2relative to this basis: [p(x)] B=[:

a)The point Q is approximately (5.91863, 5.94188).

b) The directional derivative of f at P in the direction of v is approximately 2 sqrt (24).

c)  The exact value of the directional derivative of f at P in the direction of v is 2sqrt(24).

The exact value of the directional derivative of f at P in the direction of v is 2sqrt(24)

(a) To find point Q, we need to move a distance of 0.1 in the direction of vector v = ⟨-1, 1⟩ from the point P = (6, 6). Let Q = (x, y) be the desired point. Then we have:

Q = P + t v

where t is the distance we need to travel in the direction of v to reach Q. Since the length of v is sqrt(2), we have t = 0.1 / sqrt(2). Substituting the given values, we get:

Q = (6, 6) + (0.1/sqrt(2)) ⟨-1, 1⟩ = (5.91863, 5.94188) (rounded to five decimal places)

Therefore, the point Q is approximately (5.91863, 5.94188).

(b) To approximate the directional derivative of f at P in the direction of v, we use the formula:

fv ≈ (∇f(P) · v)

where ∇f(P) is the gradient of f at P. We have:

∇f(x,y) = ⟨1/2sqrt(x+3y), 3/2sqrt(x+3y)⟩

∇f(6,6) = ⟨1/2sqrt(6+3(6)), 3/2sqrt(6+3(6))⟩ = ⟨1/2sqrt(24), 3/2sqrt(24)⟩

v = ⟨-1, 1⟩

Therefore, we have:

fv ≈ (∇f(P) · v) = ⟨1/2sqrt(24), 3/2sqrt(24)⟩ · ⟨-1, 1⟩

fv ≈ -sqrt(24)/2 + 3sqrt(24)/2

fv ≈ 2sqrt(24)

Therefore, the directional derivative of f at P in the direction of v is approximately 2sqrt(24).

(c) The exact value of the directional derivative of f at P in the direction of v is given by the formula:

fv = (∇f(P) · v)

Using the values of ∇f(P) and v from part (b), we get:

fv = ⟨1/2sqrt(24), 3/2sqrt(24)⟩ · ⟨-1, 1⟩

fv = -sqrt(24)/2 + 3sqrt(24)/2

fv = 2sqrt(24)

Therefore, the exact value of the directional derivative of f at P in the direction of v is 2sqrt(24).

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The measure of the angle BC will be ∠BC = 72°.

A chord of a circle is a straight line segment that connects two points on the circle's circumference. The length of a chord is the distance between the two points.

The portion of a straight line that joins two points on a circle is known as the chord's length. It is the longest distance between the two points on the circle. The radius of the circle and the separation between the two spots on the circle determine the chord's length.

The angle BC will be calculated as,

∠BC = ( ∠BDC / 180 ) x π

∠BC = (72 / 180 ) x π

∠BC = 72°

Therefore, the value of angle BC will be 72°.

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Answer: About 43 sq: ft. About 86 sq.

Step-by-step explanation: