Let be a fixed vector in and vector be a solution to where Q is a m*n matrix.
Prove every solution to the equation is in the form?

Answer:

the last one

Step-by-step explanation:

The congruent angles, ∠ACD and ∠BCD formed by the angle bisector \(\overline{DC}\) the congruent segments \(\overline{AC}\) and \(\overline{BC}\), and the segment \(\overline{DC}\) (congruent to itself, indicates that ΔACD ≅ ΔBCD by SAS

The SAS (Acronym for Side-Angle-Side) congruency rule states that two triangles, A and B are congruent if two sides and an included angle of the triangle A are congruent to the two sides and an included angle of the other triangle B.

The specified information are;

The bisector of ∠ACB = \(\overline{DC}\)

Segment \(\overline{AC}\) ≅ \(\overline{BC}\)

Required; to prove that ΔACD ≅ ΔBCD

The two-column method can be used to prove the congruency of the triangles as follows;

Step     \({}\)             Statement            \({}\)        Reasons

1.  \({}\)                    \(\overline{DC}\) bisects ∠ACB          Given

                 \({}\)      \(\overline{AC}\) ≅ \(\overline{BC}\)

2.  \({}\)                  ∠ACD ≅ ∠BCD              Definition of bisected angles

3.                \({}\)    \(\overline{CD}\) ≅ \(\overline{CD}\)                       Reflexive property of congruency

4. \({}\)                   ΔACD ≅ ΔBCD               SAS congruency rule

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The percentage recovery of paraffin wax for the given flow rate in the final extract is equal to -92%.

Assumptions are,

The process operates under ideal conditions, meaning no losses or inefficiencies occur during the extraction and separation process.

The composition and properties of the wax paper, paraffin wax, kerosene, and other materials remain constant throughout the process.

The sludge underflow from each thickener consists only of insoluble wax paper and the overflow solvent, with no other impurities present.

The solvent is able to fully dissolve and extract the paraffin wax from the wax paper.

The separation stages (agitated vessel and thickener) are efficient in separating the paraffin wax from the solvent and wax paper.

There are no side reactions or chemical changes occurring during the process.

Now let's calculate the percentage recovery of paraffin wax in the final extract.

Flow rate of wax paper = 320 kg/hr

Flow rate of paraffin wax to be leached =200 kg/hr

Flow rate of kerosene = 480 kg/hr

Sludge underflow contains 3.2 kg of overflow per kg of insoluble wax paper

First, let's calculate the flow rate of insoluble wax paper in the sludge underflow,

Flow rate of insoluble wax paper

= Flow rate of wax paper - Flow rate of paraffin wax

= 320 kg/hr - 200 kg/hr

= 120 kg/hr

Next, calculate the flow rate of the overflow solvent in the sludge underflow,

Flow rate of overflow solvent

= Flow rate of insoluble wax paper × Overflow per kg of insoluble wax paper

= 120 kg/hr × 3.2 kg/kg

= 384 kg/hr

Now, let's calculate the flow rate of paraffin wax in the final extract,

Flow rate of paraffin wax in the final extract

= Flow rate of paraffin wax - Flow rate of overflow solvent

= 200 kg/hr - 384 kg/hr

= -184 kg/hr (negative value indicates loss)

Since the flow rate of paraffin wax in the final extract is negative,

it means there is a loss in the process and the recovery of paraffin wax is less than 100%.

To calculate the percentage recovery of paraffin wax in the final extract, consider the initial amount of paraffin wax to be leached,

Percentage recovery

= (Flow rate of paraffin wax in the final extract / Flow rate of paraffin wax to be leached) × 100

= (-184 kg/hr / 200 kg/hr) × 100

= -92%

This negative value indicates a loss in the process,

and the amount of paraffin wax obtained in the final extract is lower than the initial amount to be leached.

Therefore, the percentage recovery of paraffin wax in the final extract is -92%.

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Answer:

Step-by-step explanation:

The zeros are the values of x for which f(x) = 0.

f(x) = (x-3)(2x-8) = 0

x = 3, 4

Answer:

maximum score =75

minimum=0

Step-by-step explanation:

This means that the function g(t) has a critical point at t = 0. If the Hessian of the function f(x, y) at ⟨a, b⟩ is positive-definite, then the Hessian of the function g(t) at t = 0 is also positive-definite. This means that g(t) has a relative minimum at t = 0.

Let a function f(x, y) has a relative minimum at ⟨a, b⟩. The function g(t) :

= f(a + t, b + t) has a relative minimum at t

= 0. We are asked to explain why this is the case.The function g(t) :

= f(a + t, b + t) is a composition of the function f(x, y) with the vector-valued function h(t) :

= (a + t, b + t). By the chain rule, we have:g'(t)

= (∇f)(h(t)) · h'(t),where ∇f is the gradient of the function f(x, y).Note that h(0)

= ⟨a, b⟩, so that g(0)

= f(a, b). Since f(x, y) has a relative minimum at ⟨a, b⟩, we have ∇f(a, b)

= 0. Thus, g'(0)

= 0, since h'(0)

= (1, 1) and ∇f(a, b) · (1, 1)

= 0. This means that the function g(t) has a critical point at t

= 0. If the Hessian of the function f(x, y) at ⟨a, b⟩ is positive-definite, then the Hessian of the function g(t) at t

= 0 is also positive-definite. This means that g(t) has a relative minimum at t

= 0.

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The reference angle of -200° is 20°. If you think about it, the terminal arm will be in Q2. That means that it has a reference angle of 200-180 = 20°

The sequence is a geometric sequence with a common ratio of 7. Since, the common ratio is greater than 1, the sequence diverges.

A convergent series is a series whose partial sums tend to a specific number, also called a limit. A divergent series is a series whose partial sums, by contrast, don't approach a limit. Divergent series typically go to ∞, go to −∞, or don't approach one specific number.

The given sequence is 1/117659, 1/16807, 1/2401...

Here, common ratio is

1/16807 ÷ 1/117659

= 1/16807 × 117659/1

= 117659/16807

= 7

Now, 1/2401 ÷ 1/16807

= 1/2401 × 16807/1

= 16807/2401

= 7

Hence, the sequence is a geometric sequence with a common ratio of 7. Since, the common ratio is greater than 1, the sequence diverges.

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The total cost to operate 5 fluorescent 40w light bulbs for the months of April and May if they are on for 3 hours/day is given as follows:

$1,830.

The total cost is obtained applying the proportions in the context of the problem.

The number of kilowatt hours is given as follows:

Hence:

200 x 61 x 3 = 36,600 kwh.

The cost is of $0.05 per kwh, hence the total cost is given as follows:

0.05 x 36600 = $1,830.

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Answer:

yes

Step-by-step explanation:

If you divide one into the other and the answer is 1 then they are of the same ratio.

Answer:

yes they are

when you multiply 4 x 2 you get 8 and when you divide 3 x 2 you get 6

Answer:

\( \frac{9}{2} \)

Step-by-step explanation:

\( \frac{15}{2} \div \frac{5}{3} = \frac{15}{2} \times \frac{3}{5} = \frac{3}{2} \times \frac{3}{1} = \frac{9}{2} \)

The approximate volume of the sphere is 6.01 ft³.

A sphere is simply a three-dimensional geometric object that is perfectly symmetrical in all directions.

The volume of a sphere is expressed as:

Volume =  (4/3)πr³

Where r is the radius of the sphere and π is the mathematical constant pi (approximately equal to 3.14).

Given that the surface area of the sphere is 16 square feet.

First, we determine the radius r:

Surface area = 4πr²

Hence

16 = 4πr²

Dividing both sides by 4π, we get:

r² = 16/4πr

r = √( 16/4πr )

r = 1.128 ft

Plugging in the value of r that we just found, we get:

Volume =  (4/3)πr³

Volume =  (4/3) × 3.14 × (1.128 ft)³

Volume =  6.01 ft³

Therefore, teh volume is 6.01 ft³.

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The treatment sum of squares measures the variability of the sample treatment means around the overall mean.

A mean can be defined as a ratio of the sum of the total number in a data set (population) to the frequency of the data set.

This ultimately implies that, a mean score is an average score and it can be calculated by dividing the total number of scores obtained by total number of samples in a data set (population).

In ANOVA, the following sum of squares can be used to measure the variation about the mean:

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In a tesselation of the plane with congruent equilateral triangles, each vertex is shared by six triangles.

This can be visualized by imagining a single equilateral triangle as the center, surrounded by six other equilateral triangles that share a common vertex with the central triangle. Therefore, in this type of tesselation, each vertex is surrounded by six neighboring triangles.

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\(~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^{-m}\implies a^{n-m} \\\\[-0.35em] ~\dotfill\\\\ (4)^{\frac{-4}{2}} \implies (4)^{-2}\implies 4^{-2}\implies \cfrac{1}{4^2}\implies \cfrac{1}{16}\)

To find the vector equation and the parametric equations of the line through a given point that is perpendicular to two given vectors, let's assume the given point is P(x₀, y₀, z₀), and the two given vectors are u and w.

First, let's find a vector that is perpendicular to both u and w. We can achieve this by taking the cross product of u and w.

Let v = u x w (cross product of u and w)

Now, we have a vector v that is perpendicular to both u and w. To find the vector equation of the line through point P that is perpendicular to u and w, we can write:

r = P + tv

where r is the position vector of any point on the line, t is a scalar parameter, and v is the vector that is perpendicular to u and w.

To obtain the parametric equations, we can break down the vector equation into three component equations. Let's assume:

P(x₀, y₀, z₀)

v = (a, b, c)

r = (x, y, z)

The vector equation can be written as:

x = x₀ + at

y = y₀ + bt

z = z₀ + ct

These are the parametric equations of the line through point P that is perpendicular to u and w, where t is the parameter.

Remember to substitute the values of the given point P, as well as the components of vector v, in order to have the specific equations for your given situation.

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Tanui initially had 2 hectares of potatoes, and after adding 1 more hectare, the total area became 3 hectares. The percentage increase in the area under potatoes is 50%.

The initial area of potatoes is 2 hectares. After adding 1 more hectare, the total area becomes 2 + 1 = 3 hectares.

To calculate the percentage increase, we need to find the difference between the final and initial areas. In this case, it is 3 hectares - 2 hectares = 1 hectare.

Next, we divide the difference by the initial area: 1 hectare / 2 hectares = 0.5.

To express this as a percentage, we multiply the result by 100: 0.5 * 100 = 50.

Therefore, the percentage increase in the area under potatoes is 50%

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Answer:

Step-by-step explanation:

we have the diameter =60 cm, the radius is 60/2=30 cm

A circle is 3.14*r^2=2827 cm^2

Answer: C. 2827 cm2

Step-by-step explanation:

The area of a circle is calculated by pir^2; r = the radius

In the diagram, the diameter is provided. The radius is half of the diameter. 60/2 = 30, so the radius is 30 cm.

Plug these values into the formula

A = 3.14*30^2

A = 2827.43, which would round to 2827 cm2

The function f(x)=x²−4  f(2+x) simplifies to x² + 4x.

To determine f(2+x), we substitute (2+x) into the function f(x) = x² - 4:

f(2+x) = (2+x)² - 4

Now, let's expand and simplify the expression:

f(2+x) = (2+x)(2+x) - 4

= (2+x)(2) + (2+x)(x) - 4

= 4 + 2x + 2x + x² - 4

= x² + 4x + 4 - 4

= x² + 4x

Therefore, f(2+x) simplifies to x² + 4x.

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Answer:

B

Step-by-step explanation:

The slope of this line is 2/3 because it has a point at (0,0) and another at (3,2). This means that that is its constant of proportionality.

Answer:

B) 2/3

Step-by-step explanation:

Rise / run. Same thing as slope.

Hope this helps. Pls give brainliest.

Combining like terms is an essential skill in Algebra 1. It involves simplifying algebraic expressions by adding or subtracting terms that have the same variables and exponents. The goal is to simplify the expression by reducing it to its simplest form.

To combine like terms, we need to follow a few steps. Here is a step-by-step guide:

Identify the like terms: Look at the expression and find the terms that have the same variables and exponents. For example, in the expression 3x + 5x - 2x, all three terms have the variable x with an exponent of 1, so they are like terms.

Add or subtract the coefficients: Once you have identified the like terms, add or subtract their coefficients. The coefficient is the number that is multiplied by the variable. For example, in the expression 3x + 5x - 2x, the coefficients are 3, 5, and -2. Adding them together gives us 6x.

Write the simplified expression: After combining the like terms, write the simplified expression. In the example above, the simplified expression is 6x.

Here is another example:

4a + 2b - 3a + b

In this expression, the like terms are 4a and -3a (both have the variable a). We can subtract the coefficients to get a + 2b. The final simplified expression is a + 2b.

It is important to note that we can only combine terms that have the same variables and exponents. For example, we cannot combine 3x^2 and 4x because they have different exponents.

In summary, combining like terms is a basic skill in Algebra 1. It involves identifying terms that have the same variables and exponents, adding or subtracting their coefficients, and writing the simplified expression. This skill is essential for solving equations, simplifying expressions, and graphing functions.

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The 90% confidence interval for the mean capacitance of a certain type of capacitor with a sample size is calculated below:Given information:Sample size = LargeSample of capacitors = Certain typeConfidence Interval for the mean capacitance (μf) = (0.206, 0.242)Confidence Level = 90%As the sample size is large, we can use the z-distribution to compute the confidence interval.

The z-value at a 90% confidence level is 1.645.Therefore, the margin of error (E) is given byE = (zα/2) * (σ/√n)where zα/2 is the z-value at α/2 level of significanceσ is the standard deviationn is the sample sizeSubstituting the given values we get,E = 1.645 * (0.242 - 0.206) / √n.

Since the standard deviation (σ) is not given, we have to use the given confidence interval to calculate it.σ = (0.242 - 0.206) / (2 * 1.645)σ = 0.01848Substitute σ and n in the formula for E to get,E = 1.645 * 0.01848 / √nTherefore, the 90% confidence interval is given byμ ± E = (0.224 - 0.206) to (0.224 + 0.206) = (0.206, 0.242)Hence, the 90% confidence interval for the mean capacitance is (0.206, 0.242).

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The average value of the function f(x) = 6x² on the interval [0, 2] is 8.

To find the average value of a function on an interval, we need to calculate the integral of the function over that interval and then divide it by the length of the interval.

In this case, the function is f(x) = 6x² and the interval is [0, 2].

To find the integral of f(x), we integrate 6x² with respect to x:

∫ 6x² dx = 2x³ + C

Next, we evaluate the integral over the interval [0, 2]:

∫[0,2] 6x² dx = [2x³ + C] from 0 to 2

= (2(2)³ + C) - (2(0)³ + C)

= 16 + C - C

= 16

The length of the interval [0, 2] is 2 - 0 = 2.

Finally, we calculate the average value by dividing the integral by the length of the interval:

Average value = (Integral) / (Length of interval) = 16 / 2 = 8

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Answer:

Fraction of the garden used =  \(\frac{5}{8}\)

Step-by-step explanation:

The fifth graders to plant a butterfly garden.

We are given that

Clover grows in ½ of the garden

Daisies grow in ¼ of the garden

Coneflowers grow in ⅛ of the garden

milkweed grows in ⅛ of the garden

The fifth graders notice that the butterflies land on the clover and milkweed.

We are asked to find out the fraction of the garden that the butterflies used.

Fraction of the garden used = clover + milkweed

Fraction of the garden used =  \(\frac{1}{2} + \frac{1}{8}\)

The LCM of 2 and 8 is 8

Fraction of the garden used =  \(\frac{4 +1}{8}\)

Fraction of the garden used =  \(\frac{5}{8}\)

Therefore, the butterflies used \(\frac{5}{8}\) fraction of the garden.

Answer:

105

Step-by-step explanation:

1,000 divided by 3.5% is 35

35 x 3 is 105

Since you are just asking for just the interest and not the total the answer is 105

The interest that Amy should pay is $105.

Given that,

Based on the above information, the calculation is as follows:

\(= 1000 \times 3.5\% \times 3\)

= 105

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Answer:

118degrees

Step-by-step explanation:

Assuming we are given the following and m<k and ,<N lies on the same straight line, hence;

m<K = 62 degrees

n<N = ?

Since are on the same straight line, hence;

m<N + m<K = 180

m<N + 62 = 180

m<N = 180 - 62

m<N  = 118

Hence the measure of m<N is 118degrees

'

,

Answer: The number 6 came up 7 times out of a total of 40 trials.

Step-by-step explanation:

The experimental probability is the ratio of the number of favorable outcomes and the total outcomes in an event.

Given: The experimental probability of getting a 6 on a number cube is \(\dfrac7{40}\) .

That means, favorable outcomes= 7x, Total outcomes= 40x

So the true statement for the event would be.

The number 6 came up 7 times out of a total of 40 trials.

Answer:

B

Step-by-step explanation: