If the distance between A (0,4) and B(3,a) is 5 units then Find the value of a.​

The discrepancy between the column spaces of a matrix's echelon form U and its original form A, C(U) and C(A) respectively, originates from how column space is defined.

Column space is purported as the set of all linear combinations of the columns in the matrix. Nonetheless, despite elementary row operations modifying the linear independence of the matrix, it does not alter its column space.

Subsequently, therby resulting to C(U) being a subspace of C(A). This concept can be exemplified via an example: When we employ 3x3 matrix with linearly independent columns, then its echelon form U may only display two linearly independent columns; thus increasingly shrinking the dimension of the transformed matrix's column space.

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the speed of the train in km/h is 40.32 whereas the speed of the train in m/s is 11.2.

(i) To find the speed of the train in km/h, we can use the formula:

speed = distance/time

Here, the distance travelled by the train is 16.8 km, and the time taken is 25 minutes, which is equivalent to 0.4167 hours (since 1 hour = 60 minutes). Substituting these values in the formula, we get:

speed = 16.8 km/0.4167 hours

speed = 40.32 km/h (rounded to two decimal places)

Therefore, the speed of the train in km/h is 40.32.

(ii) To find the speed of the train in m/s, we can convert the speed in km/h to m/s by multiplying by 1000/3600 (since 1 km/h = 1000 m/3600 s). Using the speed of 40.32 km/h from part (i), we get:

speed = 40.32 km/h * 1000 m/km / 3600 s/h

speed = 11.2 m/s (rounded to one decimal place)

Therefore, the speed of the train in m/s is 11.2.

If the speed limit is 55 km/h, we cannot directly determine the distance travelled by the train without additional information. The distance travelled by the train would depend on the time it took to travel at the given speed limit.

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Answer:

(-1, 2)

Step-by-step explanation:

When an equation is formatted like this, you can just reverse the operators (in this case +1 and -2) and you'll get your coordinates.

The area of ACD based on the information given will be 54 units².

The area of a trapezoid is calculated as follows:

(base1 + base2) height / 2 = Area

where base1 and base2 are the lengths of the trapezoid's two parallel sides, and height is the perpendicular distance between them.

So, assuming you have the measurements for base1, base2, and height, you can use the formula above to compute the area of the trapezoid.

Area = 1/2 × (10 + 8) × 6

Area = 1/2 × 18 × 6

Area = 54 units ²

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1. Required profit function is P(x)= -0.5x² + 60x + 6850

2. The domain of P(x) is 20 - 4√205 ≤ x ≤ 170.

3. P(70) = 2650 and P(80) = 2600.

The company should choose to produce 70 items as it results in a higher profit.

4. The leads to a decrease in profit, which is why the company makes less profit when producing 10 more units.

What is profit function?

A profit function is a mathematical formula that describes the relationship between the level of production or sales and the resulting profit of a business. The profit function takes into account the costs of production, including fixed costs and variable costs, and the revenue generated by the sale of goods or services.

1. The profit function P(x) is given by subtracting the cost function C(x) from the revenue function R(x):

P(x) = R(x) - C(x)

= [−0.5(x−120)²+7,200] - [50x+250]

= -0.5x² + 60x + 6850

2. The domain of P(x) is the set of all possible values of x for which the profit function P(x) makes sense. Since P(x) involves subtracting the cost function from the revenue function, it only makes sense to calculate P(x) when the revenue generated from selling x items is greater than or equal to the cost of producing x items. Therefore, the domain of P(x) is the set of all x such that R(x) ≥ C(x),

R(x) ≥ C(x)

-0.5(x−120)²+7,200 ≥ 50x+250

-0.5x²+60x+6850 ≥ 50x+250

-0.5x²+10x+6600 ≥ 0

Solving for x, we get:

x ≤ 20 - 4√205 or x ≥ 20 + 4√205

Since the maximum capacity of the company is 170 items, the domain of P(x) is the intersection of the above solution and the maximum capacity, i.e. x ≤ 170. Therefore, the domain of P(x) is 20 - 4√205 ≤ x ≤ 170.

3. To find the profit when producing 70 items, substitute x = 70 into the profit function P(70) = -0.5(70)² + 60(70) + 6850

= 2650

To find the profit when producing 80 items, substitute x = 80 into the profit function P(80) = -0.5(80)² + 60(80) + 6850

= 2600

Therefore, the company should choose to produce 70 items as it results in a higher profit.

4. The profit function P(x) is a quadratic function with a negative leading coefficient (-0.5), which means that it opens downwards. This implies that the profit function reaches its maximum value at the vertex of the parabola, which occurs at x = -b/2a, where a = -0.5 and b = 60. Therefore, the vertex occurs at x = -60/-1 = 60. This means that the maximum profit occurs at x = 60.

When the company produces 10 more units (i.e. x increases by 10), it moves away from the optimal production level of 60 and closer to the maximum capacity of 170. As a result, the cost of production increases while the revenue generated from selling those extra units decreases due to diminishing returns. This leads to a decrease in profit, which is why the company makes less profit when producing 10 more units.

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The better statistics to use to compare the distributions includes mean and standard deviation. The Option C is correct.

Mean represents the central tendency of the distribution, while standard deviation reflects the spread of the data around the mean. Comparing mean & standard deviation of 2 or more distributions can provide insights into similarities and differences.

If the means of two distributions are similar, it suggests the data points in each distribution are centered around similar value. But if standard deviations are also similar, it indicates that the spread of the data points around the mean is also similar.

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Answer:

f(x) =

0 if -1 < x < 0,

1 if 0 < x < 3,

2 otherwise

Answer:

Plot the points on the graphing calculator, and then determine the linear regression function: y = -1.16786x + 8.82857

You may obtain a slightly different linear regression equation.

1) -1 (g)

2) 8 (not one of the choices)

3) graph (i)

4) approximately -1/strong negative trend (not one of the choices)

5) y = -x + 8 (h)

This expression [tex]r_{i-j}[/tex] describes the distance between two points i and j in a geometric object

In the context of symmetry and rotations, [tex]r_{i-j}[/tex] typically refers to the distance between two points i and j in a geometric object, such as a crystal lattice or a molecule.

It is a vector that points from point i to point j, and its magnitude is the distance between the two points.

The distance vector [tex]r_{i-j}[/tex] is also used to describe the position of a point in the crystal lattice relative to the rotation axis.

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Answer:

s = 10√3

Step-by-step explanation:

This is a 30°-60°-90° right triangle, so s, the length of the longer leg, is √3 times the length of the shorter leg. Since the length of the shorter leg is 10, s = 10√3.

In this case, renting the huge van would be a more cost-effective option because it would run you $289 as opposed to $354.65 for the minivan.

To compare the costs of renting the minivan and the large van, we need to calculate the total cost for each option.

For the minivan, the cost per day is $24.95 and the cost per mile is $0.15. So, for 7 days and 1,200 miles, the total cost would be:

Total cost = 7($24.95) + 1,200($0.15)

= $354.65

For the large van, the cost is a flat rate of $289 per week with no additional charge for mileage. So, for 7 days and 1,200 miles, the total cost would be:

Total cost = $289

Therefore, renting the large van would be a better buy in this scenario, as it would cost $289 compared to $354.65 for the minivan.

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The recursive equation for this table is given as follows:

f(n) = 3f(n - 1) + 1.

To obtain the recursive equation, we must obtain a pattern between a term and the previous term.

For this problem, we have that each term is the previous term multiplied by 3 and added by one.

Hence the recursive equation for this table is given as follows:

f(n) = 3f(n - 1) + 1.

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Hence, the value of CosФ + Cos (2π+Ф) + Cos (4π+Ф) is 0.6 .

The cosine function One of the fundamental trigonometry functions is cos x as the others being the co-secant, cotangent, secant, sine, and tangent. Let theta represent the angle that can be calculated by rotating the unit circle's arc anticlockwise from the x-axis. Cos theta is the arc endpoint of  horizontal coordinate after that.

trigonometric functions are used in an expression or equation, trigonometric Identities are helpful. For every value of a variable appearing on both sides of an equation, a trigonometric identity is true. Certain trigonometric functions such as sine, cosine, and tangent of one or more angles are involved geometrically in these identities.Three main are  trigonometry functions are sine, cosine, and tangent, while the other three are cotangent, secant, and co-secant. All six trigonometric functions serve as the foundation for the trigonometric identities.

Given, CosФ = 0.2,

thus we can use the trigonometric identity:

cos(2 n π+Ф) = cosФ

where any integer n is used. Therefore:

Cos(2 π+ Ф) = CosФ

cos (4π+Ф) = cosФ

These values, put into the given expression

than we get,

cos Ф+ cos (2π+Ф) + cos (4π+Ф) = cos Ф+ cosФ + cosФ.

= 3 cosФ

= 3×0.2

= 0.6

So,CosФ + Cos (2π+Ф) + Cos (4π+Ф) = 0.6

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Answer:

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

The range of the given function has one restriction, its denominator can't be zero, hence:

Therefore the function can get any value but zero:

The matching answer choice is D.

Linear function is f(x) = mx + b.

Reciprocal of a function f(x)  is 1/f(x).

So the reciprocal of linear function has a form of f(x) = 1 / (mx + b).

The only answer choice in same form is D.

Substitute x = - 5/3 into function to get:

This is undefined value, therefore it represents the vertical asymptote.

The function gets close to - ∞ when x → - 5/3 from left and gets close to +∞ when x → - 5/3 from right (see attached graph).

Therefore the correct choice is B.

Answer:

43

Step-by-step explanation:

The distance formula is

[tex] \sqrt{{(x2 - x1)}^{2} + (y2 - y1)^{2} } [/tex]

So

[tex] \sqrt{(25 + 18) ^{2} + ( - 7 + 7)^{2} } \\ = \sqrt{ {43}^{2} + 0 {}^{2} } = 43[/tex]

Answer:

Step-by-step explanation:

The distance between the two given points can be found using the formula for the distance between two points in a coordinate plane,

which is d = √((x₂ - x₁)² + (y₂ - y₁)²). In this case, the x-coordinates of both points are the same (-7), so we only need to calculate the difference between their y-coordinates: d = √((0)² + (25 - (-18))²) = √(43²) = 43 units. Therefore, the distance between the two points located at (-7, -18) and (-7, 25) is 43 units

Answer:

If I am not mistaken a=3;b=10;c=23;d=39

Step-by-step explanation:

A loan of $595 borrowed on June 15 and repaid on Dec 17 at an exact interest rate of 6%. The exact time between the two dates is 170 days, and the maturity value is $611.69 rounded to the nearest cent.

Using the Days in a year table, we can find the exact time between June 15 and December 17 as follows

June has 30 days in the table and July through November have 31 days each. December has 17 days until the loan is repaid. Therefore, the exact time is

30 + 31 + 31 + 30 + 31 + 17 = 170 days

Next, we can calculate the interest using the formula

interest = principal x rate x time / 365

where the principal is $595, the rate is 6%, and the time is 170 days.

Substituting these values, we get

interest = 595 x 0.06 x 170 / 365

interest = $16.69

Therefore, the exact interest is $16.69.

Finally, we can calculate the maturity value by adding the principal and the interest

maturity value = principal + interest

maturity value = $595 + $16.69

maturity value = $611.69

Therefore, the maturity value is $611.69 rounded to the nearest cent.

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The distance between Eilat and Jerusalem is 181.4 miles.

To convert kilometers to miles, we need to multiply the number of kilometers by 0.621371, which is the conversion factor from kilometers to miles. Therefore, to convert the distance between Eilat and Jerusalem from kilometers to miles, we can use the following formula:

distance in miles = distance in kilometers × 0.621371

Substituting the given distance of 292 kilometers into the formula, we get:

distance in miles = 292 km × 0.621371 = 181.417852 miles

Rounding this answer to the nearest tenth, we get:

distance in miles ≈ 181.4 miles

Therefore, the distance between Eilat and Jerusalem is approximately 181.4 miles.

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Step-by-step explanation:

(3h - 5)(5h + 4) =   3h * 5h    +   3h * 4     - 5*5h    - 5 *4

                              =    15h^2        - 13h    -20

[tex](3h-5)(5h+4)[/tex]

[tex]=(3h+-5)(5h+4)[/tex]

[tex]=(3h)(5h)+(3h)(4)+(-5)(5h)+(-5)(4)[/tex]

[tex]=15h^2+12h-25h-20[/tex]

Answer:

Along with proof of (a.) and (d.), (b.) Power tower: one level is a, (k + 1) levels is a raised to the power of a power tower with k levels, (c.) A(2, n) <= 2 ↑↑ n for all positive integers n, where ↑↑ denotes power tower notation.

The idea of a fully computable function that is not primitive recursive is illustrated by the recursively constructed mathematical function known as the Ackermann function. Since m and n are non-negative integers, it is commonly written as A(m, n).

a.) Prove using regular induction that [tex]A(1, n) \leq 2^n[/tex] for all positive integers n:

Base Case: For n = 1, A(1, 1) = 2, which is equal to [tex]2^1[/tex].

Inductive Hypothesis: Assume that [tex]A(1, k) \leq 2^k[/tex] for some positive integer k.

Inductive Step: We need to show that [tex]A(1, k + 1) \leq 2^{(k + 1)}[/tex]. Using the recursive definition of A(m, n), we have [tex]A(1, k + 1) = A(0, A(1, k)) = 2^{(A(1, k))}\leq 2^{(2^k)}[/tex] (by inductive hypothesis)[tex]< = 2^{(2^{(k + 1)})}[/tex].

Therefore, by regular induction, we have proved that [tex]A(1, n) \leq 2^n[/tex] for all positive integers n.

b.) A power tower with one level is defined as a, and a power tower with (k + 1) levels is defined as a raised to the power of a power tower with k levels.

c.) [tex]A(2, n) \leq 2[/tex] ↑↑ n for all positive integers n, where ↑↑ denotes power tower notation.

d.) The recursive definition of a triple arrow-up notation for power towers is:

a ↑↑↑ 1 = a (base case)

a ↑↑↑ (k + 1) = a ↑↑ (a ↑↑↑ k) (recursive step)

This definition states that a triple arrow-up notation with one level is equal to the base value "a", and a triple arrow-up notation with (k + 1) levels is equal to "a" raised to the power of a triple arrow-up notation with "k" levels.

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Complete Question: ( Refer to image)

The factored form of        [tex]8x^{2} 4 - 27y^{6}[/tex] 8x²4 - 27y⁶

To factor this expression, that it is in the form of a difference of two squares.

Specifically, 8x²4 is equal to (2x²)³ and 27y⁶ is equal to (3y²)³.

the formula for the difference between two cubes, states that:

              [tex]a^3 - b^3 = (a - b)(a^2 + ab + b^2)[/tex]  (a³ - b³ = (a - b)(a² + ab + b²)

Substituting [tex]a = 2x^2[/tex] a = 2x² and [tex]b = 3y^2[/tex] b = 3y², gives

[tex]8x^24 - 27y^6 = (2x^2)^3 - (3y^2)^3= (2x^2 - 3y^2)(4x^2 + 6x^2y^2 + 9y^4)[/tex]

8x²4 - 27y⁶ = (2x²)³ - (3y²)³ = (2x² - 3y²)(4x⁴ + 6x²y² + 9y⁴)

Therefore, the factored form of [tex]8x^{2} 4 - 27y^{6}[/tex] 8x²4 - 27y⁶ is [tex](2x^2 - 3y^2)(4x^4 + 6x^2y^2 + 9y^4).[/tex](2x² - 3y²)(4x⁴ + 6x²y² + 9y⁴).

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Answer: 5 5/8 more feet of snow in December.

Step-by-step explanation: 5 7/8 also equals 47/8, and we need convert 1/4 to something over 8 so we can subtract the two. 1/4, but multiply both sides by 2 is 2/8. 47/8-2/8=45/8. 45/8 as a mixed fraction is 5 5/8.

2. 28 goldfish was what was given away before noon than in the afternoon

3. 108 goldfish were given away all day

4. Yes the owner have enough goldfish for the entire day

5. 16 goldfish were left over

2. Before noon, gold fish = 30 + 36 + 40 = 106

after noon gold fish = 28 + 30 + 10 + 2 + 8 = 78

diference = 106 - 78

= 28

3. Gold fish given away all day = 106 + 78

= 184

4. The owner had enough fish because the total fist he had was 200 and the amount that was given away is 184

5. The left over fish = 200 - 184

= 16

6. The owner was nervous because the customers were increasing between 9 to 11 and she had given away more than half in the first three hours

7. A reason no one came at noon could be because a resstaurant was having a lunch special

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Answer: To find the area of the ring-shaped path, we need to subtract the area of the inner circle (the pool) from the area of the outer circle. The radius of the pool is half the diameter, so it is 11 yards. The radius of the outer circle is the sum of the radius of the pool and the width of the path, so it is 11 + 5 = 16 yards.

The area of the pool is:

A_pool = pi * r^2 = pi * 11^2 ≈ 380.13 square yards

The area of the outer circle is:

A_outer = pi * R^2 = pi * 16^2 ≈ 804.25 square yards

The area of the ring-shaped path is:

A_path = A_outer - A_pool ≈ 804.25 - 380.13 ≈ 424.12 square yards

Since one gallon of coating can cover 5 square yards, we need:

Gallons = A_path / 5 ≈ 424.12 / 5 ≈ 84.82

Therefore, we need approximately 85 gallons of coating to cover the ring-shaped path.

Step-by-step explanation:

Step-by-step explanation:

6 or 7.

that means either exactly 6 or exactly 7 out of the 10 board the flight.

these are non-depending (or non-overlapping) scenarios. so, for this "or" remained we can simply add the 2 individual probabilities P(6) and P(7).

P(6) is the probability of exactly 6 passengers out of 10 will board the flight.

because of the 75% certainty overall we can conclude that the probabilty for each booked passenger to board is 0.75 (75% means 75 out of 100 = 0.75).

and the probabilty to not board is

1 - 0.75 = 0.25

so, for a specific group of exactly 6 booked passengers out of 10 their probabilty to board is

0.75⁶ × 0.25⁴ = 0.000695229...

6 passengers board, 4 do not board.

but this is for only one possible grouping of 6 passengers out of 10.

since the sequence does not matter, and we have no repetitions (every passenger counts as 1), we have combinations :

C(10, 6) = 10! / (6! × (10-6)!) = 10!/(6!×4!) = 10×9×8×7/4! =

= 5×3×2×7 = 210

therefore,

P(6) = 210 × 0.75⁶ × 0.25⁴ = 0.145998001...

and for a specific group of exactly 7 booked passengers out of 10 their probabilty to board is

0.75⁷ × 0.25³ = 0.002085686...

7 passengers board, 3 do not board.

but this is for only one possible grouping of 7 passengers out of 10.

since the sequence does not matter, and we have no repetitions (every passenger counts as 1), we have combinations :

C(10, 7) = 10! / (7! × (10-7)!) = 10!/(7!×3!) = 10×9×8/3! =

= 5×3×8 = 120

therefore,

P(7) = 120 × 0.75⁷ × 0.25³ = 0.250282288...

so, the probability that either 6 or 7 booked passengers are actually boarding is

P(6) + P(7) = 0.396280289... ≈ 0.396

Answer:

9

Step-by-step explanation:

All you have to do is replace the letters with numbers

= ( x - y) + z

= ( 4 - 5) +  10

= -1 + 10

= 10 - 1

= 9

Where the above information about right triangles are given, the asnswer is

A. x = 3, y = 2 (Option A)

The hypotenuse and the length of one leg of one right triangle must be equal to the hypotenuse and corresponding length of the one leg of the other ∆ for both triangles to be equal by the HL Congruence Theorem.

Thus, let's find x and y by setting the corresponding lengths of the two right ∆s equal to each other.

Therefore:

x = y + 1 ----› eqn. 1

2x + 3 = 3y + 3 ----› eqn. 2

Substitute x = (y + 1) into eqn. 2, and solve for y.

2x + 3 = 3y + 3 ----› eqn. 2

2(y + 1) + 3 = 3y + 3

2y + 2 + 3 = 3y + 3

2y + 5 = 3y + 3

Collect like terms

2y - 3y = -5 + 3

-y = -2

Divide both sides by -1

y = 2

Substitute y = 2 into eqn. 1.

x = y + 1 ----› eqn. 1

x = 2 + 1

x = 3

Thus,

x = 3, and y = 2 (Option A)

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Full Question:

See attached image

The probabilities will be :

P(none) = 0.018

P(at least one) =  0.982

P(at most one) = 0.187

Pretty Betty share  is 15%, the probability that any one household is tuned to the show will be:  0.15.

To find the probability that none of the households are tuned to Pretty Betty will be:

P(none) = 0.85¹¹

            ≈ 0.018

So, to find the probability that at least one household is tuned to Pretty Betty,  i use the complement rule as well as subtract the probability of none of the households being tuned to the show from 1:

P(at least one)

= 1 - P(none)

≈ 1 - 0.018

≈ 0.982

So, to find the probability that at most one household is tuned to Pretty Betty:

P(at most one) = P(none) + P(one)

= 0.85^11 + 11(0.15)(0.85^10)

≈ 0.187

Hence, the probabilities will be :

P(none) = 0.018

P(at least one) =  0.982

P(at most one) = 0.187

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