consider the following series.∑[infinity] xn 12n n = 11. Find the values of x for which the series converges. Answer (in interval notation):2. Find the sum of the series for those values of x. Write the formula in terms of x. Sum:

However, it does not appear to be a familiar series, and finding an exact sum may be difficult or impossible.

To determine the convergence of the series, we can use the limit comparison test.

Let's compare the given series with the series 1/n^2, since both of them have positive terms.

lim n→∞ ln[(n^2+8)/(3n^2+1)] / (1/n^2)

= lim n→∞ [ln(n^2+8) - ln(3n^2+1)] / (1/n^2)

= lim n→∞ [(2n/(n^2+8)) - (6n/(3n^2+1))] * n^2

= lim n→∞ [2/(1+(8/n^2)) - 6/(3+(1/n^2))]

The limit of this expression can be evaluated by dividing the numerator and denominator by n^2, which gives:

lim n→∞ [2/(1+(8/n^2)) - 6/(3+(1/n^2))]

= lim n→∞ [2/(n^2/n^2+(8/n^2)) - 6/(n^2/n^2+(3/n^2))]

= lim n→∞ [2/(1+(8/n^2)) - 6/(1+(3/n^2))] * (1/n^2)

Now we can see that the limit is of the form (finite number) * (1/n^2), which goes to zero as n approaches infinity. Therefore, by the limit comparison test, since the series 1/n^2 is convergent (p-series with p=2), the given series is also convergent.

Since the series is convergent, we can find its sum using any appropriate method. However, it does not appear to be a familiar series, and finding an exact sum may be difficult or impossible.

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To rewrite the given parametric equations as Cartesian equations, we need to eliminate the parameter t. For the first equation, we get the Cartesian equation y = (3/2)x - (5/4). For the second equation, we get the Cartesian equation y = ln(x^2).

For the first equation x(t) = 3t - 2, y(t) = 5t^2, we need to eliminate t to get the Cartesian equation. Solving for t in terms of x, we get t = (x + 2)/3. Substituting this value in the equation for y, we get y = 5((x+2)/3)^2. Simplifying this, we get y = (3/2)x - (5/4).

For the second equation x(t) = e^(2t), y(t) = e^(4t), we need to eliminate t to get the Cartesian equation. Taking the natural logarithm of both sides of the equation for y, we get ln(y) = 4t.

Solving for t, we get t = ln(y)/4. Substituting this value in the equation for x, we get x = e^(2(ln(y)/4)), which simplifies to x = y^(1/2). Therefore, the Cartesian equation for this parametric equation is y = ln(x^2).

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part a.   the total bus fare cost for Mrs Themba's family per week is R337.

part b. The cost per bus trip per child is found to be around R9.50.

Cost is described as an amount that has to be paid or spent to buy or obtain something.

The cost of a weekly bus pass from Blue Downs to Rondebosch =  R109.

We then find the , the total cost per week for Mrs. Theba's family

1 x R109+ 2 x R114  = R337

part b.

The cost of a weekly bus pass from Bellville to Cape Town =  R95.

We find the cost per bus trip per child as:

R95 /  10 = R9.50

So in conclusion,  the cost per bus trip per child is R9.50.

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For the given equation,there is no solution that satisfies the initial condition x(0) = 7.

What is equation?

An equation is a statement that shows the equality between two expressions, typically separated by an equals sign. Equations are used to represent relationships between variables or quantities, and solving an equation involves finding the values of the variables that satisfy the equality.

We start by separating the variables:

[tex]dx/dt = x^2/125\\\\(125/x^2) dx = dt[/tex]

Integrating both sides gives:

-125/x = t + C

where C is the constant of integration. To find C, we use the initial condition x(0) = 7:

-125/7 = 0 + C

C = -125/7

Substituting this back into our equation, we have:

-125/x = t - 125/7

Solving for x, we get:

x = 125/(t - 125/7)

This is the general solution to the differential equation. To find the particular solution that satisfies the initial condition x(0) = 7, we substitute t = 0 and x = 7 into the general solution:

7 = 125/(0 - 125/7)

7 = 125/( - 125/7)

7 = -7

This is a contradiction, which means that there is no solution that satisfies the initial condition x(0) = 7.

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C-Child ticket

4C-Adult tickets

6.10C+9.90(4C)=1188.20

6.10C+39.60C=1188.20

45.70C=1188.20

Divide both sides by 45.70 to get C.

C=26

This means 26 child tickets were sold that day.

Let's check our answer:

26×6.10=158.60

26×4=104 (Adult tickets)

104×9.90=1029.60

158.60+1029.60=1188.20

Therefore, 26 child tickets were sold on Friday.

The calculated total weight of the fruit is 50 2/5 pounds

From the question, we have the following parameters that can be used in our computation:

This means that

Banana = 6 * Apple

So, we have

Banana = 6 * 7 2/10 pounds.

Evaluate the products

Banana = 43 2/10 pounds.

So, the total weight is

total weight = apple + banana

This gives

total weight = 7 2/10 + 43 2/10

Evaluate the sum

total weight = 50 4/10

Simplify

total weight = 50 2/5

Hence, the total weight is 50 2/5 pounds

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Answer:

x=8inches²/6+x

Step-by-step explanation:

Area Photo: 1/2 area entire ad

Area photo: 4inch*2inch= 8inch²

Area entire ad=8inch²*2=16inch²

as you can see in the photo, we can use the data to our advantage.

The area left that's not the photo is (4*x) +(x*x)+(2*x).

so that leaves us with the equation 8inch²=4x+2x+x²

8inch²=6x+x²

x(6+x)=8inch²

x=(8inch²)/(6+x)

we have shown that n is divisible by d, which means that d | n.

What is Derivation ?

Derivation is a mathematical technique used to find the rate at which a function changes. In other words, it is a method for calculating the instantaneous rate of change of a function at a particular point. Derivation is an important tool in calculus, and it has a wide range of applications in fields such as physics, engineering, and economics.

We are given that dc | nc, which means that there exists an integer k such that nc = k(dc).

We need to show that d | n, which means that there exists an integer m such that n = md.

We can start by dividing both sides of the given equation nc = k(dc) by c:

n = k(d)

Since d and k are integers, their product k(d) is also an integer, which means that n is an integer.

Therefore, we have shown that n is divisible by d, which means that d | n.

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The data described is a combination of both nominal and ordinal data. In the first round of the spelling bee is nominal data, and in the last round of the spelling bee is ordinal data.

The first round of the spelling bee is nominal data as it is a categorical variable that describes the order in which the participants ranked in that round. The first round did not have a specific rank or order, it was simply a categorical grouping. However, the last round of the spelling bee is an example of ordinal data as it is based on a specific order or rank. The participant came in first place in the last round, which is an example of ordinal data.

Ordinal data is a type of data that represents a specific order or ranking, while nominal data represents a categorical grouping. In the context of the spelling bee, the first round is an example of nominal data because it is simply a grouping of participants who performed in a certain way. In contrast, the last round of the spelling bee is an example of ordinal data because it involves a specific ranking or order that the participants achieved.

As it involves both categorical groupings and specific rankings. Understanding the difference between nominal and ordinal data is important when analyzing and interpreting data, as it can affect the statistical methods and techniques used to analyze the data.

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The derivative of f(x)=4/3x⁵/4 at the point x=4 is 5.06.

This is found by using the power rule, which states that the derivative of xⁿ is n*xⁿ⁻¹. In this case, we have n=5/4, so the derivative is (5/4)*(4/3)*4¹/⁴ = 5.06.

The power rule is a common method for finding derivatives of functions with powers. It states that if f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. Using this rule, we can find the derivative of f(x)=4/3x⁵/4 by first multiplying the constant 4/3 by the power of x, which gives 4/3 * (5/4)x⁽⁵/⁴⁻¹⁾. Simplifying this expression gives us the derivative f'(x) = (5/3)x¹/⁴.

To find the value of the derivative at x=4, we simply plug in x=4 to get f'(4) = (5/3)4¹/⁴ = 5.06 (rounded to two decimal places). This tells us the rate of change of the function at the specific point x=4.

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The correct answer to convert 20 inches to feet using dimensional analysis is B. Divide 20 inches by 1ft/12 in.

To convert 20 inches to feet using dimensional analysis, we need to set up a conversion factor that relates inches to feet. We know that there are 12 inches in one foot, so we can write the conversion factor as 1 ft / 12 in. We want to cancel out the units of inches, so we can write 20 inches as 20 in / 1. Then, we can multiply 20 in / 1 by our conversion factor, making sure that the units cancel out appropriately:

20 in / 1 × 1 ft / 12 in = 20/12 ft

Simplifying, we get:

20 in / 1 × 1 ft / 12 in = 1.67 ft

Therefore, 20 inches is equal to 1.67 feet when using dimensional analysis and dividing by the conversion factor of 1ft/12in.

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The 70th percentile of current measurement is 11.05 milliamperes.

To find the 70th percentile of the current measurement, we need to find the value of the current measurement that separates the lowest 70% of measurements from the highest 30% of measurements.

We can use a standard normal distribution table or a calculator to find the z-score that corresponds to the 70th percentile, which is 0.5244.

Then we can use the formula:

x = μ + zσ

where x is the value of the current measurement, μ is the mean of the distribution, σ is the standard deviation, and z is the z-score corresponding to the 70th percentile.

Plugging in the values, we get:

x = 10 + 0.5244(2) = 11.05

Therefore, the 70th percentile of current measurement is 11.05 milliamperes.

So, the answer is 11.05.

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The derivative of the cross product u(t) × v(t) with respect to t is given by:

d/dt [u(t) × v(t)] = [d/dt u(t)] × v(t) + u(t) × [d/dt v(t)]

Using the given functions, we have:

d/dt [u(t) × v(t)] = [leftangle0.gif 8cos(8t), 8sin(8t), 1 rightangle0.gif] × [t, cos(8t), sin(8t)] + [sin(8t), cos(8t), t] × [leftangle0.gif -8sin(8t), 8cos(8t), 0 rightangle0.gif]

Simplifying this expression, we get:

d/dt [u(t) × v(t)] = [8t, -8sin^2(8t), 8cos^2(8t)] + [8sin(8t), 8cos^2(8t), -8sin^2(8t)]

Therefore, the derivative of the cross product is:

d/dt [u(t) × v(t)] = [8t + 8sin(8t), 8cos^2(8t) - 8sin^2(8t), 8cos^2(8t) - 8sin^2(8t)]

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Based on this, we can conclude that an increase in price will result in a decrease in total revenue, since the increase in price will be offset by a larger decrease in quantity demanded

To find the elasticity of demand, we need to calculate the derivative of q with respect to p multiplied by the ratio of p to q.

Taking the derivative of the demand function with respect to p, we get:

q + 100 = -p/q

Multiplying both sides by p/q, we get:

p/q * q + 100p/q = -p

Simplifying, we get:

p/q = -100/(q^2 - p)

When p = $200, we can substitute this value into the equation to get:

200/q = -100/(q^2 - 200)

Solving for q, we get:

q = 50

So at a price of $200, the quantity demanded is 50 units. To find the elasticity, we need to calculate:

E = (dq/dp) * (p/q)

Taking the derivative of the demand function with respect to p, we get:

dq/dp = -1/q^2

Substituting p = $200 and q = 50, we get:

dq/dp = -1/2500

Substituting into the formula for elasticity, we get:

E = (-1/2500) [tex]\times[/tex] (200/50) = -0.16

Since the elasticity is negative, we know that demand is inversely related to price, meaning that as the price increases, the quantity demanded will decrease.

Since the elasticity is greater than 1 in absolute value, we know that demand is elastic, meaning that a change in price will result in a relatively larger change in quantity demanded.

Based on this, we can conclude that an increase in price will result in a decrease in total revenue, since the increase in price will be offset by a larger decrease in quantity demanded.

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The successive crests and troughs on the periodic function graph indicates that the period is 6.0 seconds, therefore;

A periodic function is a function that repeats the same values of the output variable at regular intervals.

The coordinates of the points on the periodic function graph are; (2.5, 2900), (5.5, 1100), (8.5, 2900), and (11.5, 1100)

The period is the time it takes to complete a cycle of the periodic function, which is the time between successive crests or troughs.

The crests and troughs in the graph are;

Crest; (2.5, 2900), (8.5, 2900)

Trough; (5.5, 1100), (11.5, 1100)

The period, which is the time between successive crests and troughs are therefore;

Period, T = 8.5 - 2.5 = 11.5 - 5.5 = 6.0

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The individual linearizations of f(x) and f'(x) at x0 are combined to obtain the linearization of f(x) at x0.

To find the linearization of a function f(x) at a point x0, we use the following formula:

L(x) = f(x0) + f'(x0)(x - x0)

where f'(x0) represents the derivative of f(x) evaluated at x0.

The linearization of f(x) at x0 is an approximation of the function near x0, where the approximation is a linear function. It is related to the individual linearizations of f(x) and f'(x) at x0 in the following way:

The linearization of f(x) at x0 is a linear function that approximates f(x) near x0. It can be seen as the "best" linear approximation of f(x) near x0.

The linearization of f'(x) at x0 is a constant value that represents the slope of the tangent line to f(x) at x0. This constant value is also known as the instantaneous rate of change of f(x) at x0.

The linearization of f(x) at x0 can be obtained by combining the constant value f(x0) and the linear function f'(x0)(x - x0). The linear function represents the change in f(x) as x moves away from x0, while the constant value f(x0) represents the value of f(x) at x0.

Therefore, the individual linearizations of f(x) and f'(x) at x0 are combined to obtain the linearization of f(x) at x0.

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we should guess 4 or 5 as the first digit to maximize our chance of being correct.

The graph below shows the approximate probabilities for n = 1 to 10.

we find that this occurs when n is approximately 65.

a) Since the digits are independent and uniformly distributed, the expected value of each digit is 4.5.

Therefore, we should guess 4 or 5 as the first digit to maximize our chance of being correct.

b) For n = 1, there is a 10% chance of guessing correctly. For n = 2, there are 100 possible two-digit numbers, and only 11 of them have an average of 4 or 5 (04, 05, 13, 14, 22, 23, 31, 32, 40, 41, and 50).

Therefore, the chance of guessing correctly is 11/100 or 11%. For larger values of n, we can approximate the probability using the central limit theorem. The distribution of Dn approaches a normal distribution with mean 4.5 and standard deviation sqrt(8.25/n). Therefore, the probability of guessing correctly can be approximated by the area under the normal curve between 3.5 and 5.5. The graph below shows the approximate probabilities for n = 1 to 10.

c) We want to find the smallest value of n such that the probability of guessing correctly is at least 0.99. From the central limit theorem, we know that the probability of guessing correctly is approximately normal with mean 4.5 and standard deviation sqrt(8.25/n).

Therefore, we want to find the smallest value of n such that the area under the normal curve to the right of 5.5 is at least 0.01. Using a standard normal table or calculator, we find that this occurs when n is approximately 65.

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The correct answer is gcd(Greatest common divisor)  and lcm(least common multiple) of 1000 and 625.

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Answer:

b

Step-by-step explanation:

Answer:

Step-by-step explanation:

The point (6, 1) is not a solution to the system.

(6, 1) lays on the dotted line.

The area of the region that lies inside the first curve and outside the second curve. r = 3 − 3 sin(), r = 3 is 9π/2.

The two polar curves given are:

r1 = 3 - 3sin(θ)

r2 = 3

The region that lies inside the first curve and outside the second curve is the region bounded by these two curves. To find the area of this region, we need to integrate the area element over the region.

The area element in polar coordinates is given by dA = r dr dθ. Therefore, the area of the region can be computed as:

A = ∫θ1^θ2 ∫r2^r1 r dr dθ

where θ1 and θ2 are the angles at which the two curves intersect.

To find the intersection points, we set the two equations equal to each other:

3 - 3sin(θ) = 3

Simplifying, we get:

sin(θ) = 0

which implies that θ = 0 or θ = π.

Therefore, the integral becomes:

A = ∫0^π ∫3-3sin(θ)^3 r dr dθ

= ∫0^π [(1/2)r^2]_3-3sin(θ) dθ

= (1/2) ∫0^π (9 - 18sin(θ) + 9sin(θ)^2) dθ

= (1/2) [9θ + 6cos(θ) - 9sin(θ)]_0^π

= 9π/2

Therefore, the area of the region that lies inside the first curve and outside the second curve is 9π/2.

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The partial fraction decomposition for this integrand will have the form:
x2 − 2x − 1 = A/(x-1) + B/(x-1)

Once we find A and B, we can substitute them back into the original equation and integrate each term separately. This will allow us to evaluate the integral of x2 − 2x − 1 using the partial fraction decomposition.

To perform a partial fraction decomposition on the integrand x2 − 2x − 1, we first need to factor the denominator into linear factors. The quadratic x2 − 2x − 1 can be factored as (x-1)(x-1), which means we have a repeated linear factor of (x-1).

To decompose this, we need to write it in the form of a fraction with a numerator and denominator. The numerator will have a constant term for each repeated linear factor, and the denominator will be the product of each linear factor.

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The correct answer is 9.

To find the limit of lim x→0 sin(8x) tan(9x), we can use the fact that sin(8x)/x → 8 as x → 0 and tan(9x)/x → 9 as x → 0. Therefore, the limit becomes:

lim x→0 sin(8x) tan(9x) = lim x→0 (8x/8) (9x/tan(9x))
= 72 lim x→0 (sin(8x)/(8x)) (x/tan(9x))

Using L'Hospital's rule on the first factor, we get:
lim x→0 sin(8x)/(8x) = lim x→0 (cos(8x))/8 = 1/8

Using L'Hospital's rule again on the second factor, we get:
lim x→0 x/tan(9x) = lim x→0 (1/cos^2(9x)) = 1

Therefore, the overall limit is:
lim x→0 sin(8x) tan(9x) = 72 (1/8) (1) = 9

So, the correct answer is 9.

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The bias of mu1 is 0, the variance of mu1 is 1/2, and the mean squared error of mu1 is 1/2.

To find the bias, variance, and mean squared error of mu1:

We can use the following formulas:
Bias = E[mu1] - mu
Variance = Var[mu1]
MSE = E[(mu1 - mu)^2]
First, let's find E[mu1]:
E[mu1] = E[(x1 + x2)/2]
Since x1 and x2 are independent, their expected values are equal to mu:
E[x1] = E[x2] = mu
Therefore, E[mu1] = E[(x1 + x2)/2] = (E[x1] + E[x2])/2 = mu.
Next, let's find Var[mu1]:
Var[mu1] = Var[(x1 + x2)/2]
Since x1 and x2 are independent, their variances are both equal to (sigma)^2 = 1:
Var[x1] = Var[x2] = (sigma)^2 = 1
Therefore, Var[mu1] = Var[(x1 + x2)/2] = (1/4)*Var[x1] + (1/4)*Var[x2] = 1/2.
Finally, let's find MSE:
MSE = E[(mu1 - mu)^2]
= E[(x1 + x2)/2 - mu)^2]
= E[((x1 - mu) + (x2 - mu))/2]^2
= E[(x1 - mu)^2 + 2(x1 - mu)(x2 - mu) + (x2 - mu)^2]/4
= (E[(x1 - mu)^2] + E[(x2 - mu)^2] + 2E[(x1 - mu)(x2 - mu)])/4
= (Var[x1] + Var[x2] + 2Cov[x1,x2])/4
Since x1 and x2 are independent, their covariance is 0:
Cov[x1,x2] = E[(x1 - mu)(x2 - mu)]

= E[x1x2 - mu(x1 + x2) + mu^2]

= E[x1]E[x2] - mu(E[x1] + E[x2]) + mu^2
= mu^2 - mu^2 - mu^2 + mu^2 = 0
Therefore, MSE = (Var[x1] + Var[x2])/4 = 1/2.
In summary, the bias of mu1 is 0, the variance of mu1 is 1/2, and the mean squared error of mu1 is 1/2.

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The values of dy and δy for the function y=e3x 5x given x=0 and δx=dx=0.03 are:
dy = 0.6
δy = 0.6

To compute the values of dy and δy for the function y=e3x 5x given x=0 and δx=dx=0.03, we need to use the formula for the total differential of a function:

dy = (∂y/∂x)dx

where ∂y/∂x is the partial derivative of y with respect to x.

In this case, we have:

y = e3x 5x

∂y/∂x = 3e3x 5x + e3x 5

At x=0, this becomes:

∂y/∂x = 3(1) 5 + (1) 5 = 20

So, we can now calculate dy:

dy = (∂y/∂x)dx = (20)(0.03) = 0.6

This means that when x changes by 0.03, y changes by 0.6.

To calculate δy, we need to use the formula:

δy = |(∂y/∂x)δx|

where δx is the uncertainty in x.

In this case, we have:

δy = |(20)(0.03)| = 0.6

So, the uncertainty in y is also 0.6.

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Answer:

64

Step-by-step explanation:

Number of outcomes = number of choices per question ^ number of questions

In this case, the number of choices per question is 4 and the number of questions is 3. Plugging these values into the formula, we get:

Number of outcomes = 4^3 = 64

To solve the given 1-dimensional heat equation problem, we can use the method of separation of variables. The problem is defined as follows:

Partial Differential Equation (PDE): ∂u/∂t = α^2 ∂^2u/∂x^2, for t > 0 and 0 < x < 5.

Boundary conditions:
1. u(0, t) = 0
2. u(5, t) = 0

Initial condition: u(x, 0) = f(x) = -4 sin(Tx) + 3 sin(27x), for 0 < x < 5.

To solve this problem, perform the following steps:

1. Assume a solution in the form u(x, t) = X(x)T(t).
2. Substitute this solution into the PDE and separate the variables.
3. Solve the resulting ordinary differential equations (ODEs) for X(x) and T(t) subject to the given boundary conditions.
4. Obtain the general solution by summing the product of the separated solutions X_n(x)T_n(t) with appropriate coefficients.
5. Determine the coefficients by applying the initial condition and using Fourier series representation.

Since the problem is well-posed, a unique solution exists.

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For the linear transformation T that rotates points about some line through the origin, an eigenvalue of A is 1. The eigenspace associated with this eigenvalue is the subspace of all vectors parallel to the axis of rotation.

Since T is a rotation about some line through the origin, it has an axis of rotation, which is the line through the origin that remains fixed by the transformation.

Any vector on this line will be an eigenvector of the transformation with eigenvalue 1, since it remains fixed under the transformation.

Therefore, an eigenvalue of A is 1. The eigenspace corresponding to this eigenvalue is the subspace spanned by the axis of rotation.

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An advantage of using graphs and diagrams is B. They help to visualize the problem.

Graphs and diagrams are pictorial representations of data.

Graphs represent information using lines on two or three axes such as x, y, and z.

On the other hand, diagrams show the simple pictorial representation of what a thing looks like or how it works.

Graphs are scaled while diagrams may not be scaled.

Thus, we use graphs and diagrams to visualize data and information.

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The equation of the plane with normal vector =⟨−5,2,5⟩ passing through the point =(4,1,8) in scalar form is 5x + 2y + 5z = 22.

1. Recall the equation of a plane in scalar form: Ax + By + Cz = D, where ⟨A, B, C⟩ is the normal vector of the plane, and (x, y, z) are the coordinates of any point on the plane.

2. In this case, the normal vector is given as ⟨−5, 2, 5⟩. Therefore, A = -5, B = 2, and C = 5.

3. The plane passes through the point (4, 1, 8). We can use this point to find the value of D. Substitute the point's coordinates into the equation: -5(4) + 2(1) + 5(8) = D.

4. Calculate the value of D: -20 + 2 + 40 = 22.

5. Now, we can write the equation of the plane in scalar form using the values of A, B, C, and D: -5x + 2y + 5z = 22.

So, the equation of the plane with normal vector ⟨−5, 2, 5⟩ passing through the point (4, 1, 8) in scalar form is: -5x + 2y + 5z = 22.

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cut shape into two which is triangle and a trapezium use to formulas of the identified shapes in solving the area