you bought two new pairs of shoes with the last of your money in your checking account. your next payday isn't until next friday. what is opportunity cost of these shoes?

To evaluate the integral Jx e^5x2 dx by making the substitution u = 5x^2, we need to first find the value of dx in terms of du. Taking the derivative of u with respect to x, we get du/dx = 10x, which can be rearranged to dx = du/10x.
Substituting this into the integral, we get:
Jx e^5x2 dx = J(u/10)e^u (du/10x)
Simplifying this expression, we get:
Jx e^5x2 dx = (1/10)Jue^u du
To evaluate this new integral, we can use integration by parts, with u = u and dv/dx = e^u. This gives us:
Jx e^5x2 dx = (1/10)(ue^u - J e^u du) + C
Resubstituting for u, we get:
Jx e^5x2 dx = (1/10)(5x^2e^5x^2 - Jx e^5x^2 dx) + C
Simplifying and rearranging, we get:
Jx e^5x2 dx = (1/20)x e^5x^2 + C

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The conclusion about this school’s absences is that there is not enough data to determine if there is a difference in absences. Therefore, the correct option is D.

Based on the data provided for the 5 students, we cannot conclusively determine whether there is a significant difference between the number of absences in Fall and Winter months. The reason is that the sample size is too small (only 5 students) to draw any strong conclusions. Therefore, there is not enough data to determine if there is a difference in absences.

To make a more accurate conclusion, the school would need to collect data from a larger sample of students and potentially conduct a statistical test, such as a paired t-test, to determine if there is a significant difference in absences between the two seasons. Hence, the correct answer is option D.

Note: The question is incomplete. The complete question probably is: a high school conducts a dependent sample experiment to test if there is a difference between the number of times students are absent in the fall and winter months. What do you conclude about this school’s absences? You have data for the following 5 students:

STUDENT FALL WINTER

1 1 0

2 2 1

3 3 0

4 2 1

5 0 1

A) There is no difference between the Fall and Winter months. B) There is a statistically significant difference between the Fall and Winter months. C) There is a difference but it is only somewhat statistically significant. D) There is not enough data to determine if there is a difference in absences.

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Alice can be expected to win approximately 5 games out of the 7 games she plays.

If historically Alice wins 70% of the time, we can expect her to win 70% of the games she plays. If Alice and Bob play 7 games of chess, we can calculate how many games Alice can be expected to win by multiplying the total number of games (7) by the probability of Alice winning (70% or 0.7):

Expected number of games Alice wins = 7 * 0.7 = 4.9

Since we cannot have a fraction of a game, we can round the expected number of games Alice wins to the nearest whole number. Therefore, Alice can be expected to win approximately 5 games out of the 7 games she plays.

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If m∠t = 3 ⋅ m∠s then m∠s is equal to 45° based using the properties of supplementary angles ( sum of their measures is 180°) and Division Property of equality.

The paragraph proof for the given two-column proof is as follows:

We are given that m∠t = 3 ⋅ m∠s. By the Linear Pair Theorem, we know that ∠t and ∠s are supplementary angles. This means that the sum of their measures is 180° (Definition of Supplementary Angles).

Therefore, we can write the equation m∠t + m∠s = 180°. Substituting the given information, we have 3 ⋅ m∠s + m∠s = 180°. Combining like terms, we get 4 ⋅ m∠s = 180°.

To find the measure of ∠s, we divide both sides of the equation by 4, resulting in m∠s = 45° (Division Property of Equality).

Thus, we have proven that m∠s is equal to 45° based on the given information and using the properties of supplementary angles and equality.

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The statistical method commonly used to analyze the clustering of personality traits is called cluster analysis.

Cluster analysis is a multivariate statistical technique that aims to identify groups or clusters of similar observations or variables based on their characteristics or measurements.

In the context of personality traits, cluster analysis can be used to explore patterns of similarity or dissimilarity among individuals based on their trait profiles.

In cluster analysis, various algorithms and distance metrics are employed to determine the grouping of observations.

The goal is to create homogeneous clusters, where individuals within the same cluster are more similar to each other compared to individuals in different clusters.

The choice of algorithm and distance metric depends on the specific objectives of the analysis and the nature of the data.

Once the cluster analysis is performed, researchers can interpret and describe the identified clusters, examine the characteristics that differentiate the clusters, and investigate the relationships between the clusters and other variables of interest.

This allows for a deeper understanding of the underlying patterns and structures within the personality trait data.

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it is shown that if events A and B are independent, then A' and B are also independent.

If events A and B are independent, it can be shown that the complement of event A (A') and event B are also independent. This means that the occurrence or non-occurrence of event A does not affect the probability of event B, and vice versa.

The proof involves using the definition of independence and the complement rule.

To show that A' and B are independent, we need to demonstrate that the probability of the intersection of A' and B is equal to the product of their individual probabilities.

Let's denote P(A) as the probability of event A, P(B) as the probability of event B, and P(A') as the probability of the complement of event A.

By the definition of independence, we have P(A ∩ B) = P(A) * P(B).

Now, using the complement rule, we know that P(A') = 1 - P(A).

Next, we can express event A' and B in terms of their intersection:

A' ∩ B = (A ∪ A') ∩ B

= (A ∩ B) ∪ (A' ∩ B)

Using the inclusion-exclusion principle, we can rewrite the above equation as:

P(A' ∩ B) = P(A ∪ A') + P(B) - P(A ∩ B)

Since A and A' are mutually exclusive, we have P(A ∪ A') = P(A) + P(A') = P(A) + 1 - P(A) = 1.

Substituting these values into the equation, we get:

P(A' ∩ B) = 1 + P(B) - P(A ∩ B) - P(A ∩ B)

= 1 + P(B) - 2 × P(A ∩ B)

Now, we can rewrite P(A ∩ B) using the independence of A and B:

P(A' ∩ B) = 1 + P(B) - 2 × P(A) × P(B)

= 1 + P(B) × (1 - 2 × P(A))

Since P(A) and P(B) are independent events, we have P(B) * P(A) = P(A ∩ B), which simplifies the equation to:

P(A' ∩ B) = 1 - P(A') * P(B)

Therefore, we have shown that if events A and B are independent, then A' and B are also independent.

The probability of the intersection of A' and B is equal to the product of their individual probabilities, which confirms their independence.

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The profit percentage is 6 % and the selling price of the book is A = ₹883.33

Given data ,

To find out how much the bookseller should sell the book for to gain a profit of ₹50, we can use the concept of profit percentage.

Profit percentage is calculated as (Profit / Cost Price) * 100.

Given that the bookseller gains a profit of ₹30 by selling the book for ₹500, we can calculate the profit percentage:

Profit Percentage = (Profit / Cost Price) * 100

Profit Percentage = (₹30 / ₹500) * 100

Profit Percentage = 6%

To calculate the selling price for a profit of ₹50, we need to determine what percentage of the cost price is ₹50. We can use the same profit percentage to calculate the selling price:

Profit Percentage = (Profit / Cost Price) * 100

6% = (₹50 / Cost Price) * 100

Solving for the cost price:

Cost Price = (₹50 * 100) / 6

Cost Price = ₹833.33

Now, to calculate the selling price for a profit of ₹50, we add the cost price and the desired profit:

Selling Price = Cost Price + Desired Profit

Selling Price = ₹833.33 + ₹50

Selling Price = ₹883.33

Hence , the bookseller should sell the book for ₹883.33 to gain a profit of ₹50

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The protractor can be used to obtain the angle of a table.

Place the protractor so that one of its straight edges coincides with a table edge. Make sure the protractor's center point is directly above the vertex (corner) of the desired angle.

Make that the protractor's baseline, which is typically denoted by a zero or 180 degrees, is parallel to the edge of the table. This step is essential for measuring angles accurately.

Locate the degree markings on the protractor's scale. Find the degree at which the scale intersects with the other edge of the table. The angle that the two table edges make is represented by this number.

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Answer:

c = bd ⁄ ad+b

The conditional pdf of y given x is f(y|x) = 2/x for 0 < y < x < 1.

To find the conditional probability density function (pdf) of y given x, we need to compute the conditional probability density function f(y|x). The conditional pdf represents the probability distribution of the variable y when the value of x is known.

The joint pdf is given as f(x, y) = 6y for 0 < y < x < 1.

To find f(y|x), we need to fix the value of x and normalize the joint pdf with respect to y.

For a fixed value of x, the range of y is from 0 to x. Therefore, we can express the conditional pdf as:

f(y|x) = f(x, y) / ∫[0,x] f(x, y) dy

Substituting the given joint pdf:

f(y|x) = (6y) / ∫[0,x] 6y dy

Integrating with respect to y:

f(y|x) = (6y) / [3y^2] evaluated from 0 to x

= (6x) / (3x^2)

= 2 / x

Therefore, the conditional pdf of y given x is f(y|x) = 2/x for 0 < y < x < 1.

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The midpoint of the line segment XY is S

The midpoint refers to the center or middle position of a line. To obtain the midpoint here, we take the sum of the distances and divide by 2 .

XP = 2

PQ = 3

QR = 4

RS = 3

ST = 6

TY = 6

Sum of the points :

2+3+4+3+6+6 = 24

The midpoint = 24/2 = 12

The point where 12 lies on the segment is the point S

Hence, the midpoint is S

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All matrices X that diagonalize the matrix A are invertible matrices with eigenvectors of A as their columns.

To find the eigenvectors and matrices that diagonalize matrix A, let's consider the given matrix A = [1].

To find the eigenvectors, we need to solve the equation (A - λI)v = 0, where λ is the eigenvalue and I is the identity matrix.

For matrix A = [1], subtracting λI gives:

[A - λI] = [1 - λ]

Setting the determinant of [A - λI] equal to zero will give us the eigenvalues. In this case, since [A - λI] is a scalar matrix, the determinant is simply (1 - λ).

Setting (1 - λ) = 0, we find λ = 1.

To find the eigenvector corresponding to this eigenvalue, we substitute λ = 1 back into the equation (A - λI)v = 0:

[(1 - 1)]v = [0]v = 0

Here, v is a vector in the null space of the matrix A - λI, which means any non-zero vector can be an eigenvector.

Since we have only one eigenvalue (λ = 1) with infinite possible eigenvectors, we can construct matrices X that diagonalize A by choosing any invertible matrix X that has the eigenvectors of A as its columns.

In summary, for matrix A = [1], the eigenvalue is λ = 1, and any non-zero vector can be an eigenvector. Therefore, any invertible matrix X with the eigenvectors of A as its columns can be used to diagonalize A.

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1. The line integral ∫C xy dx + y dy over the unit circle is zero.

2. The line integral ∫C (ln x + y) dx - x² dy over the given rectangle is zero.

3. The value of the line integral y dx + x dy for a simple closed curve C is equal to twice the area enclosed by C.

4. The line integral οf the vectοr field F = (x³, x) arοund the unit square οriented clοckwise is 1

5. ∫C F · dS = 0.

6. F = (P, Q) = (x + E, x + y + D).

1. To evaluate the line integral ∫C xy dx + y dy, where C is the unit circle oriented counterclockwise, we can use Green's Theorem. Green's Theorem states that the line integral of a vector field F = (P, Q) around a closed curve C is equal to the double integral of the curl of F over the region D bounded by C.

In this case, we have F = (xy, y), so the curl of F is given by ∇ × F = (∂Q/∂x - ∂P/∂y) = (1 - 1) = 0.

Since the curl of F is zero, the double integral of the curl over any region D is also zero. Therefore, the line integral ∫C xy dx + y dy over the unit circle is zero.

2. To evaluate the line integral ∫C (ln x + y) dx - x² dy over the rectangle in the xy-plane with vertices at (1, 1), (3, 1), (1, 4), and (3, 4), we can again use Green's Theorem.

First, we calculate the partial derivatives of the components of the vector field F = (ln x + y, -x²):

∂P/∂y = 1

∂Q/∂x = 1

Then, we can calculate the line integral using Green's Theorem:

∫C (ln x + y) dx - x² dy = ∬D (∂Q/∂x - ∂P/∂y) dA

The region D is the rectangle bounded by the vertices (1, 1), (3, 1), (1, 4), and (3, 4).

∬D (∂Q/∂x - ∂P/∂y) dA = ∬D (1 - 1) dA = ∬D 0 dA = 0

Therefore, the line integral ∫C (ln x + y) dx - x² dy over the given rectangle is zero.

3. If C is a simple closed curve, the value of the line integral y dx + x dy is equal to twice the area enclosed by the curve C. This result follows from Green's Theorem, which states that the line integral of the vector field F = (P, Q) over a closed curve C is equal to the double integral of the curl of F over the region enclosed by C.

Since the vector field F = (y, x) has a curl equal to 1, applying Green's Theorem gives:

∫C y dx + x dy = ∬D (∂Q/∂x - ∂P/∂y) dA = ∬D 1 dA = Area(D)

Therefore, the value of the line integral y dx + x dy for a simple closed curve C is equal to twice the area enclosed by C.

4. To evaluate the line integral of the vector field F = (x³, x) around the unit square oriented clockwise, we can use Green's Theorem.

The curl of F is given by ∇ × F = (∂Q/∂x - ∂P/∂y) = (1 - 3x²) in this case.

Since the unit square is a simple closed curve, the line integral of F around the square is equal to the double integral of the curl of F over the region enclosed by the square.

The region enclosed by the unit square can be represented as D: 0 ≤ x ≤ 1, 0 ≤ y

Using Green's Theοrem, we have:

∫C F · dS = ∬D (∂Q/∂x - ∂P/∂y) dA = ∬D (1 - 3x²) dA

Integrating with respect tο x first, we get:

∫C F · dS = ∫[0,1] ∫[0,1] (1 - 3x²) dy dx

Evaluating the inner integral:

∫[0,1] (1 - 3x²) dy = y|0 tο 1 - 3x² = 1 - 3x²

Nοw, integrating with respect tο x:

∫C F · dS = ∫[0,1] (1 - 3x²) dx = x|0 tο 1 - x³|0 tο 1 = 1 - 0 - (0 - 0) = 1

Therefοre, the line integral οf the vectοr field F = (x³, x) arοund the unit square οriented clοckwise is 1.

5. Tο evaluate the line integral οf the vectοr field F = (-y³, 2) οver the bοundary οf the regiοn A, where A is the regiοn between the circles x² + y²= 1 and x² + y² = 4, we can use Green's Theοrem.

First, we calculate the partial derivatives οf the cοmpοnents οf the vectοr field F = (-y³, 2):

∂P/∂y = -3y²

∂Q/∂x = 0

Then, we can calculate the line integral using Green's Theοrem:

∫C F · dS = ∬D (∂Q/∂x - ∂P/∂y) dA

The regiοn D is the regiοn enclοsed by the οuter circle x² + y² = 4.

Since the curl οf F is zerο (as ∂Q/∂x - ∂P/∂y = 0), the line integral arοund the bοundary οf A is zerο.

Therefοre, ∫C F · dS = 0.

6. The area abοve the x-axis and under οne arch οf the cyclοid given parametrically by x = f(t) = t - sin(t), y = g(t) = 1 - cοs(t) can be cοmputed using Green's Theοrem.

Let C be the curve defined by the cyclοid. We need tο find the vectοr field F such that its curl is equal tο 1. Then, we can evaluate the line integral οf F · dS οver C tο find the area.

Let F = (P, Q) be the vectοr field. We need tο find P and Q such that ∂Q/∂x - ∂P/∂y = 1.

In this case, we have ∂Q/∂x - ∂P/∂y = ∂Q/∂x = 1.

Integrating ∂Q/∂x with respect tο x, we get Q = x + C(y), where C(y) is an arbitrary functiοn οf y.

Taking the partial derivative οf Q with respect tο y, we have ∂Q/∂y = C'(y).

Tο satisfy the cοnditiοn that ∂Q/∂x - ∂P/∂y = 1, we need C'(y) = 1.

Integrating C'(y) = 1 with respect tο y, we get C(y) = y + D, where D is a cοnstant.

Therefοre, Q = x + y + D.

Nοw, we need tο find P. Since the curl οf F is 1, ∂P/∂x = 1.

Integrating ∂P/∂x = 1 with respect tο x, we get P = x + E, where E is a cοnstant.

Therefοre, F = (P, Q) = (x + E, x + y + D).

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The simplified expression in standard form is -143/150.

To simplify and write the given expression in standard form, follow these steps:

1. Find the least common denominator (LCD) of the fractions: The denominators are 5, 10, 15, and 25. The LCD of these numbers is 150.

2. Rewrite each fraction with the LCD as the denominator:

3/5 = (3 * 30)/(5 * 30)

= 90/150 -9/10

= (-9 * 15)/(10 * 15)

= -135/150 -11/15

= (-11 * 10)/(15 * 10)

= -110/150 2/25

= (2 * 6)/(25 * 6)

= 12/150

3. Add the fractions:

90/150 + (-135/150) + (-110/150) + 12/150

= (90 - 135 - 110 + 12)/150

= -143/150

So, the simplified expression in standard form is -143/150.

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Answer:

its either 10 or 90 but i think its 10

when running a line, in a right-triangle, from the 90° angle perpendicular to its opposite side, we will end up with three similar triangles, one Small, one Medium and a containing Large one.  Check the picture below.

[tex]\cfrac{19}{y}=\cfrac{y}{10}\implies 190=y^2\implies \sqrt{190}=y[/tex]

The value that would fill in the blank that is in the question is  2.1.

Inverse variation, also known as inverse proportionality, is a mathematical relationship between two variables in which a change in one variable leads to an opposite change in the other variable.

It's important to note that inverse variation is different from direct variation, where the two variables change in the same direction.

Given that;

y α 1/x

Then;

If y = 3 then x = 6

We can find k from;

y = k/x

k = xy

k = 3 * 6

k = 18

Then;

y = 18/8.5

y = 2.1

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Answer: 1. It would take your friend 24 minutes to catch up. Your friend would've walked 1.6 miles once they've caught up.

Step-by-step explanation: If you walk 2.5 miles per hour, you would have to divide that by 60 minutes to find out how fast you walk per minute. Which would be 0.04166666667 miles per minute, if you walked for 24 minutes, you would've completed a full mile. Your friend walks 4 miles per hour, you would want to divide that by 60 as well to find out how fast he walks per minute. Which would be 0.06666666667 miles per minute, if he walked for a total of 24 minutes, he would've walked for a total of 1.6 miles. Remember, you started 0.6 miles ahead of him, meaning after a total of 24 minutes, your friend would've caught up to you.

To find the area bounded by the polar curve r=e^−θ/12 in the given sector, we use the formula A = ∫a b 1/z,2 dθ. Here, a=π/2 and b=π, and z=e^−θ/12. So, we have A = ∫π/2 π 1/(e^−θ/6) dθ. Therefore, the area of the region bounded by the polar equation r=e^−θ/12 in the sector π/2 ≤θ≤π is 1 - e^(-π/24).

To solve this integral, we use the substitution u=-θ/12, which gives us du/dθ = -1/12, or dθ = -12du. Substituting this and e^u for e^−θ/12, we get A = ∫−π/24 0 e^u * (-12)du.
Evaluating this integral, we get A = [e^u] from −π/24 to 0 = e^0 - e^(-π/24) = 1 - e^(-π/24). So, the area of the region bounded by the polar equation r=e^−θ/12 in the given sector is 1 - e^(-π/24).
In conclusion, the area of the region bounded by the polar equation r=e^−θ/12 in the sector π/2 ≤θ≤π is 1 - e^(-π/24).

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Therefore, the volume of the solid is 16√3 cubic units.

To find the volume of the solid, we can use the method of cross-sectional areas.

The given solid has a circular base with radius 2 (since x^2 + y^2 = 4 represents a circle with radius 2). The cross sections perpendicular to the x-axis are equilateral triangles.

Let's consider a cross section at a particular x-value, denoted by x. The height of the equilateral triangle can be determined by the y-coordinate on the circle, which is given by y = √(4 - x^2).

The base of the equilateral triangle is equal to the diameter of the circle, which is 2 times the radius, or 4.

The area of an equilateral triangle is given by A = (√3 / 4) * s^2, where s is the length of a side.

In this case, the side length s is equal to the base of the equilateral triangle, which is 4.

Therefore, the area of the cross section at x is A(x) = (√3 / 4) * 4^2 = 4√3.

To find the volume of the solid, we integrate the area function A(x) over the range of x-values that spans the base of the solid.

The range of x-values can be determined by the circle equation x^2 + y^2 = 4, which implies -2 ≤ x ≤ 2.

Thus, the volume V of the solid is given by:

V = ∫[-2 to 2] A(x) dx

= ∫[-2 to 2] 4√3 dx

= 4√3 ∫[-2 to 2] dx

= 4√3 [x]_[-2 to 2]

= 4√3 * (2 - (-2))

= 4√3 * 4

= 16√3

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The correct is a. A point estimator is considered unbiased if its value is always equal to the parameter it estimates.

This means that, on average, the point estimator is accurate in estimating the population parameter. The other options are not necessarily true for an unbiased point estimator. While b and d may be desirable properties for a point estimator to have, they do not necessarily guarantee that the estimator is unbiased. Similarly, c is not a requirement for an estimator to be unbiased. The key criterion is that the estimator should provide accurate estimates of the population parameter. A point estimator is considered unbiased if the following statement is true: (d) Its sampling distribution is centered exactly at the parameter it estimates. In other words, an unbiased estimator has an expected value equal to the true parameter value, ensuring that on average, the estimates are accurate and not systematically overestimating or underestimating the true value.

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Answer:

2.6982 or 2.70$

A polynomial function of degree 3 with real coefficients that has the given zeros -2, 3, and -6 is f(x) = [tex]x^{3}[/tex] + 5[tex]x^{2}[/tex] - 42.

The zeros of a polynomial function are the values of x for which the function evaluates to zero. Since we are given the zeros -2, 3, and -6, we can express the polynomial function as the product of factors corresponding to each zero.

To find the polynomial, we use the fact that if a number is a zero of a polynomial, then the corresponding factor is equal to zero. So, for the zeros -2, 3, and -6, we have the factors (x + 2), (x - 3), and (x + 6), respectively.

Multiplying these factors together, we get:

f(x) = (x + 2)(x - 3)(x + 6)

Expanding the expression further:

f(x) = ([tex]x^{2}[/tex] - x - 6)(x + 6)

= ([tex]x^{2}[/tex] - x - 6)(x) + ([tex]x^{2}[/tex] - x - 6)(6)

= [tex]x^{3}[/tex] - [tex]x^{2}[/tex] - 6x + 6[tex]x^{2}[/tex] - 6x - 36

= [tex]x^{3}[/tex] + 5[tex]x^{2}[/tex] - 42

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To minimize costs while meeting the constraints, the optimal solution is to use 13 pounds of deluxe coffee and at least 43 pounds of regular coffee in the blend. This ensures that the minimum cost is achieved while satisfying the quantity requirements.

To solve the optimization problem step by step, let's follow the steps again

Start by considering the constraint x ≥ 13. Since we want to have at least 13 pounds of deluxe coffee, we can set x = 13.

Substitute this value of x in the first constraint: 13 + y ≥ 56. Solve for y:

13 + y ≥ 56

y ≥ 56 - 13

y ≥ 43

This means that y should be greater than or equal to 43 pounds.

We have x = 13 and y ≥ 43. These values satisfy both constraints.

Therefore, to minimize costs, we should use 13 pounds of deluxe coffee and at least 43 pounds of regular coffee.

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--The given question is incomplete, the complete question is given below "deluxe coffee is to be mixed iwth regular coffee to make at least 56 pounds of a blended coffee. the mixture must contain at least 13 pounds of deluxe coffee. deluxe coffee costs 4 perpound and regular coffee 3 per pound. how many pounds of each kind of coffee should be used to minimize costs? To attain the minimum cost, use pounds of deluxe coffee and pounds of regular coffee."--

Answer:

Hypotneuse is 2.5 cm.

Step-by-step explanation:

Hyp^2=Sum of the squares of the other two sides
a^2 = b^2+ c^2
hyp^2 = (2)^2 + (1.5)

hyp^2 = 4 + 2.25

hyp^2=6.25

Apply sq. root on both sides

hyp = 2.5 cm

In the graph, the x-axis represents the puppy's age in weeks, and the y-axis represents the puppy's weight in pounds.

In order to graph the puppy's growth based on the given equation, we need to convert it to slope-intercept form (y = mx + b), where "m" represents the slope and "b" represents the y-intercept.

Given equation: 2x - y = -2

Rearranging the equation to slope-intercept form:

y = 2x + 2

In this graph, the x-axis represents the puppy's age in weeks, and the y-axis represents the puppy's weight in pounds. The line represents the growth trend of the puppy's weight, starting from a weight of 2 pounds at 0 weeks and increasing by 2 pounds every week.

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The equation of line t in standard form is: B. 7x - y = 23.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (2 - 3)/(7 - 0)

Slope (m) = -1/7

Since the equation of this line is perpendicular to the line t, the slope is given by;

m₁ × m₂ = -1

-1/7 × m₂ = -1

m₂ = -7/-1

Slope, m₂ = 7

At data point (4, 5) and a slope of 7, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 5 = 7(x - 4)  

y = 7x - 23

7x - y = 23

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Missing information:

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

Answer:

7x - y = 23

Step-by-step explanation: 7x - y = 23**

The line with the banner will be 17 feet's for the pole to be 90 degrees with the ground .

The height of a pole is 15 feet. A line with banners is connected to the top of the pole to a point that is 8 feet from the base of the pole on the ground.

Therefore, the line with the banner to be in order for the pole to be 90 degrees angle with the ground can be found as follows:

It has to obey Pythagoras's theorem, for the pole to make 90 degrees with the ground.

Hence,

where

Therefore,

c² = 15² + 8²

c = √225 + 64

c = √289

c = 17 feet

Therefore, the length of the line should be 17 feet.

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