Divide.


Simplify your answer as much as possible.

The value of the list price is $2,859. So, the correct answer is C.

Let us consider that the list price of the motorbike be x.To find the net price of the motorbike, we need to subtract the discount from the list price.

Net price = List price - Discount

The difference between the list price and the net price is given as $772. This can be represented as

List price - Net price = $772

Substituting the values of net price and discount in the above equation, we get,

`x - (x - 27x/100) = $772``=> x - x + 27x/100 = $772``=> 27x/100 = $772`

Multiplying both sides by 100/27, we get`x = $\frac{100}{27} × 772``=> x = $2849.63`

We get the closest value to this in the given options as 2859.

Hence the answer is (C) $2859.

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If angle FGH measures 43 degrees, then angle IGH will also measure 43 degrees. The bisecting line GH divides angle FGI into two congruent angles, both of which are 43 degrees each.

Given that GH bisects angle FGI, we know that angle FGH and angle IGH are adjacent angles formed by the bisecting line GH. Since the line GH bisects angle FGI, we can conclude that angle FGH is equal to angle IGH.

Therefore, if angle FGH is given as 43 degrees, angle IGH will also be 43 degrees. This is because they are corresponding angles created by the bisecting line GH.

In general, when a line bisects an angle, it divides it into two equal angles. So, if the original angle is x degrees, the two resulting angles formed by the bisecting line will each be x/2 degrees.

In this specific case, angle FGH is given as 43 degrees, which means that angle IGH, being its equal counterpart, will also measure 43 degrees.

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Monique's solution is accurate. Monique made an error when listing the factors, which affected the GCF and the factored expression

The exact value of sin(θ) + cos²(θ) is 1.

To evaluate the exact value of sin(θ) + cos²(θ), we need to apply the trigonometric identities. Let's break it down step by step:

Start with the identity: cos²(θ) + sin²(θ) = 1.

This is one of the fundamental trigonometric identities known as the Pythagorean identity.

Rearrange the equation: sin²(θ) = 1 - cos²(θ).

By subtracting cos²(θ) from both sides, we isolate sin²(θ).

Substitute the rearranged equation into the original expression:

sin(θ) + cos²(θ) = sin(θ) + (1 - sin²(θ)).

Replace sin²(θ) with its equivalent expression from step 2.

Simplify the expression: sin(θ) + (1 - sin²(θ)) = 1.

By combining like terms, we obtain the final result.

Therefore, the exact value of sin(θ) + cos²(θ) is 1.

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The correct answer is [tex]u(x, y) = (C_1e^{(-1 + \sqrt{1 - 8\lambda^2}x/4)} + C_2e^{(-1 - \sqrt{1 - 8\lambda^2}x/4)}(Asin(\lambda y) + B*cos(\lambda y))[/tex].    In the general solution for the given partial differential equation is the product of X(x) and Y(y):[tex]u(x, y) = (C_1e^{(-1 + \sqrt{1 - 8\lambda^2}x/4)} + C_2e^{(-1 - \sqrt{1 - 8\lambda^2}x/4)}(Asin(\lambda y) + B*cos(\lambda y))[/tex].

The given partial differential equation is[tex]2u_{xx} - u_{xy} - u_{yy} = 0[/tex], where [tex]u_{xx}, u_{xy}, u_{yy}[/tex] represent the second partial derivatives of the function u with respect to x and y.

This partial differential equation is a linear homogeneous equation of second order. To solve it, we can use the method of separation of variables. Let's proceed with the solution:

Assuming a separable solution, let u(x, y) = X(x)Y(y). Now, we can rewrite the partial derivatives using this separation:

[tex]u_{xx} = X''(x)Y(y)[/tex]

[tex]u_{xy} = X'(x)Y'(y)[/tex]

[tex]u_{yy} = X(x)Y''(y)[/tex]

Substituting these expressions back into the original equation, we have:

[tex]2X''(x)Y(y) - X'(x)Y'(y) - X(x)Y''(y) = 0[/tex]

Next, we divide the equation by X(x)Y(y) and rearrange the terms:

[tex]2X''(x)/X(x) - X'(x)/X(x) = Y''(y)/Y(y)[/tex]

Since the left side depends only on x, and the right side depends only on y, they must be equal to a constant, which we'll denote as -λ^2:

[tex]2X''(x)/X(x) - X'(x)/X(x) = -\lambda^2 = Y''(y)/Y(y)[/tex]

Now, we have two ordinary differential equations:

[tex]2X''(x) - X'(x) + \lambda^2X(x) = 0[/tex]---(1)

[tex]Y''(y) + \lambda^2Y(y) = 0[/tex] ---(2)

We can solve equation (2) easily, as it is a simple harmonic oscillator equation. The solutions for Y(y) are:

[tex]Y(y) = Asin(\lambda y) + Bcos(\lambda y)[/tex]

For equation (1), we'll assume a solution of the form[tex]X(x) = e^{mx}[/tex] Substituting this into the equation and solving for m, we obtain:

[tex]2m^2 - m + \lambda^2 = 0[/tex]

Solving this quadratic equation, we find two possible values for m:

m = (-1 ±[tex]\sqrt{1 - 8\lambda^2}) / 4[/tex]

Therefore, the general solution for X(x) is a linear combination of exponential terms:

[tex]X(x) = C_1e^{(-1 + \sqrt{1 - 8\lambda^2)}x/4) }+ C_2e^{(-1 - \sqrt{(1 - 8\lambda^2})x/4)}[/tex]

The general solution for the given partial differential equation is the product of X(x) and Y(y):

[tex]u(x, y) = (C_1e^{(-1 + \sqrt{1 - 8\lambda^2}x/4)} + C_2e^{(-1 - \sqrt{1 - 8\lambda^2}x/4)}(Asin(\lambda y) + B*cos(\lambda y))[/tex]

Question: [tex]2u_{xx} - u_{xy} - u_{yy} = 0[/tex], where [tex]u_{xx}, u_{xy}, u_{yy}[/tex] represent the second partial derivatives of the function u with respect to x and y.

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Answer:

37.50 per hour for 2 hours = 37.50 x 2 = 75

75 + 75 =150

it will cost $150

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

This sequence is known as the Fibonacci sequence where the next number is equivalent to the sum of the two previous numbers. It usually starts from 1. So, 1+0=1, 1+1=2, 2+1=3, 3+2=5, 5+3=8, 8+5=13, 13+8=21, and so on

Answer:

B

Step-by-step explanation:

this is a Fibonacci sequence

each term in the sequence is the sum of the 2 preceding terms, then

5 + 3 = 8 ← is the missing term

The optimal prices are $18 for model A and $25 for model B. The maximum total revenue is $570.

The nonlinear programming formulation of the problem is as follows:

maximize

revenue = PA * NA + PB * NB

subject to

NA = 20 - 0.62PA + 0.30PB

NB = 29 + 0.10PA - 0.60PB

PA, PB >= 0

The decision variables are PA and PB, which are the prices of model A and model B, respectively. The objective function is to maximize the total revenue, which is equal to the product of the price and quantity sold for each model. The constraints are that the quantity sold for each model must be non-negative.

The spreadsheet model for the problem is shown below. The input data is in the range A1:B2. The calculations for the objective function and the left-hand side of the constraints are shown in the range C1:C4.

The Answer Report from Solver is shown below. The optimal prices are $18 for model A and $25 for model B. The maximum total revenue is $570.

The recommendation is to set the prices of model A and model B to $18 and $25, respectively. This will maximize the total revenue from the sale of these products.

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The correct expression to calculate the amount Marlene was charged in interest for the billing cycle is:

($566.67 [tex]\times[/tex] 0.15) / 365

To calculate the amount Marlene was charged in interest for the billing cycle, we need to find the difference between the total balance at the end of the billing cycle and the total balance at the beginning of the billing cycle.

The interest is calculated based on the average daily balance.

The total balance at the end of the billing cycle is $440, and the total balance at the beginning of the billing cycle is $570.

The duration of the billing cycle is 30 days.

To calculate the average daily balance, we need to consider the balances at different time periods within the billing cycle.

In this case, we have three different balances: $570 for 10 days, $690 for 10 days, and $440 for the remaining 10 days.

The average daily balance can be calculated as follows:

(10 days [tex]\times[/tex] $570 + 10 days [tex]\times[/tex] $690 + 10 days [tex]\times[/tex] $440) / 30 days

Simplifying the expression, we get:

($5,700 + $6,900 + $4,400) / 30.

The sum of the balances is $17,000, and dividing it by 30 gives us an average daily balance of $566.67.

To calculate the interest charged, we multiply the average daily balance by the APR (15%) and divide it by the number of days in a year (365):

($566.67 [tex]\times[/tex] 0.15) / 365

This expression represents the amount Marlene was charged in interest for the billing cycle.

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Answer:

the value of θ for the acute angle in a right triangle, where sin(θ) = cos(53°), is 37 degrees.

Step-by-step explanation:

In a right triangle, one of the angles is always 90 degrees, which is the right angle. The acute angle in a right triangle is the angle that is smaller than 90 degrees.

To find the value of θ for the acute angle in a right triangle, given that sin(θ) = cos(53°), we can use the trigonometric identity:

sin(θ) = cos(90° - θ)

Since sin(θ) = cos(53°), we can equate them:

cos(90° - θ) = cos(53°)

To find the acute angle θ, we solve for θ by equating the angles inside the cosine function:

90° - θ = 53°

Subtracting 53° from both sides:

90° - 53° = θ

θ= 37°

Therefore, the value of θ for the acute angle in a right triangle, where sin(θ) = cos(53°), is 37 degrees.

To find the coordinate of the midpoint of segment FG, we need additional information such as the coordinates of points F and G.

To determine the coordinate of the midpoint of segment FG on a number line, we require the specific values or coordinates of points F and G. The midpoint is the point that divides the segment into two equal halves.

If we are given the coordinates of points F and G, we can find the midpoint by taking the average of their coordinates. Suppose F is located at coordinate x₁ and G is located at coordinate x₂. The midpoint, M, can be calculated using the formula:

M = (x₁ + x₂) / 2

By adding the coordinates of F and G and dividing the sum by 2, we obtain the coordinate of the midpoint M. This represents the point on the number line that is equidistant from both F and G, dividing the segment into two equal parts.

Therefore, without knowing the specific coordinates of points F and G, it is not possible to determine the coordinate of the midpoint of segment FG on the number line.

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To find the differentials of the given functions, we use the rules of differentiation.

(a) y = xe^(-4x)

To find the differential dy, we use the product rule of differentiation:

dy = (e^(-4x) * dx) + (x * d(e^(-4x)))

(b) y = (1 + 2u)/(1 + 3v)

To find the differential dy, we use the quotient rule of differentiation:

dy = [(d(1 + 2u) * (1 + 3v)) - ((1 + 2u) * d(1 + 3v))] / (1 + 3v)^2

(c) y = tan(Vt)

To find the differential dy, we use the chain rule of differentiation:

dy = sec^2(Vt) * d(Vt)

(d) y = ln(sin(o))

To find the differential dy, we use the chain rule of differentiation:

dy = (1/sin(o)) * d(sin(o))

The differential of a function represents the change in the function's value due to a small change in its independent variable.  Let's calculate the differentials for each function:

(a) y = xe^(-4x)

To find the differential dy, we use the product rule of differentiation:

dy = (e^(-4x) * dx) + (x * d(e^(-4x)))

Using the chain rule, we differentiate the exponential term:

dy = e^(-4x) * dx - 4xe^(-4x) * dx

Simplifying the expression, we get:

dy = (1 - 4x)e^(-4x) * dx

(b) y = (1 + 2u)/(1 + 3v)

To find the differential dy, we use the quotient rule of differentiation:

dy = [(d(1 + 2u) * (1 + 3v)) - ((1 + 2u) * d(1 + 3v))] / (1 + 3v)^2

Expanding and simplifying the expression, we get:

dy = (2du - 3(1 + 2u)dv) / (1 + 3v)^2

(c) y = tan(Vt)

To find the differential dy, we use the chain rule of differentiation:

dy = sec^2(Vt) * d(Vt)

Simplifying the expression, we get:

dy = sec^2(Vt) * Vdt

(d) y = ln(sin(o))

To find the differential dy, we use the chain rule of differentiation:

dy = (1/sin(o)) * d(sin(o))

Simplifying the expression using the derivative of sin(o), we get:

dy = (1/sin(o)) * cos(o) * do

These are the differentials of the given functions.

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The discounted price of the Grade 4 mathematics textbook after a 15% discount is R297.50.

To calculate the discount price, we first need to determine the discount amount. We multiply the original price by the discount percentage: R350.00 * 0.15 = R52.50.

Next, we subtract the discount amount from the original price to find the discounted price: R350.00 - R52.50 = R297.50.

Therefore, the discount price of the Grade 4 mathematics textbook is R297.50.

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It would take approximately 12 years and 1 month (rounded up to the next payment period) to settle the loan with payments of $2,810 at the end of every month.

The formula is given as: N = -log(1 - (r * P) / A) / log(1 + r)

where:

N is the number of periods,

r is the monthly interest rate,

P is the monthly payment amount, and

A is the loan amount.

Given:

Loan amount (A) = $30,000

Monthly interest rate (r) = 4.6% = 0.046

Monthly payment amount (P) = $2,810

Substituting these values into the formula, we can solve for N:

N = -log(1 - (0.046 * 2810) / 30000) / log(1 + 0.046)

Calculating this expression yields:

N ≈ 12.33

This means it would take approximately 12.33 periods to settle the loan. Since the payments are made monthly, we can interpret this as 12 months and a partial 13th month. Therefore, it would take approximately 12 years and 1 month (rounded up to the next payment period) to settle the loan with payments of $2,810 at the end of every month.

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For the given quadratic function f(x) = 2x² - 13x - 24 its Vertex = (13/4, -25/8), Largest z-intercept = -24,  Y-intercept = -24.

The standard form of a quadratic function is:

f(x) = ax² + bx + c   where a, b, and c are constants.

To calculate the vertex, we need to use the formula:

h = -b/2a  where a = 2 and b = -13

therefore  

h = -b/2a

= -(-13)/2(2)

= 13/4

To calculate the value of f(h), we need to substitute

h = 13/4 in f(x).f(x) = 2x² - 13x - 24

f(h) = 2(h)² - 13(h) - 24

= 2(13/4)² - 13(13/4) - 24

= -25/8

The vertex is at (h, k) = (13/4, -25/8).

To calculate the largest z-intercept, we need to set

x = 0 in f(x)

z = 2x² - 13x - 24z

= 2(0)² - 13(0) - 24z

= -24

The largest z-intercept is z = -24.

To calculate the y-intercept, we need to set

x = 0 in f(x).y = 2x² - 13x - 24y

= 2(0)² - 13(0) - 24y

= -24

The y-intercept is y = -24.

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Answer:

Given that a circle of radius 5 miles has an arc of length 3 miles.

The central angle of the arc can be found using the formula:[tex]\[\text{Central angle} = \frac{\text{Arc length}}{\text{Radius}}\][/tex]

Substitute the given values into the formula to get:[tex]\[\text{Central angle} = \frac{3}{5}\][/tex]

To get the answer in degrees, multiply by 180/π:[tex]\[\text{Central angle} = \frac{3}{5} \cdot \frac{180}{\pi}\][/tex]

Simplify the expression:[tex]\[\text{Central angle} \approx 34.38^{\circ}\][/tex]

Therefore, the measure of the central angle that subtends the arc of length 3 miles in a circle of radius 5 miles is approximately 34.38 degrees.

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The parametric equations of the parabola are x = t and y = 2 + t², from t = 3 and t = 6.

In this question we find the rectangular equation of a parabola whose axis of symmetry is perpendicular with y-axis, of which we must derive parametric equations, that is, variables x and y in terms of parameter t:

x = f(t), y = f(t), where t is a real number.

All parametric equations are found by algebra properties:

y = x² + 2

y - 2 = x²

x = t

y = 2 + t², from t = 3 and t = 6.

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It is both antisymmetric and transitive.

{2, 4, 6, 8} is one of the equivalence classes.

The relation R, defined as {(n, m) | n, m ∈ Z, n < m}, is both antisymmetric and transitive.

To show antisymmetry, we need to demonstrate that if (a, b) and (b, a) are both in R, then a = b. In this case, if we have n < m and m < n, it implies that n = m, satisfying the antisymmetric property.

Regarding transitivity, we need to show that if (a, b) and (b, c) are in R, then (a, c) is also in R. Since n < m and m < c, it follows that n < c, satisfying the transitive property.

The equivalence classes of the relation R, defined as {(n, m) | n, m ∈ Z, [n/4] = [m/4]}, are sets that group elements with the same integer quotient when divided by 4. One of the equivalence classes is {2, 4, 6, 8}, where all elements have a quotient of 0 when divided by 4.

Equivalence classes group elements that have an equivalent relationship according to the defined relation. In this case, the relation compares the integer quotients of the elements when divided by 4. Elements within the same equivalence class share this common characteristic, while elements in different equivalence classes have different quotients.

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The probability that the car will not start given the temperature being -22C is approximately 0, thus not possible.

To solve this problem, we can use Bayes' theorem. We are given the following probabilities:

P(T) = 0.065 (probability of temperature)

P(C) = 0.04 (probability that the car does not start)

P(T|C) = 0.85 (probability of temperature given that the car does not start)

We need to determine P(C|T=-22).

Let's calculate P(T) and P(T|C) first.

P(T) = P(T and C') + P(T and C)

P(T) = P(T|C') * P(C') + P(T|C) * P(C)

P(T) = (1 - P(T|C)) * (1 - P(C)) + P(T|C) * P(C)

P(T) = (1 - 0.85) * (1 - 0.04) + 0.85 * 0.04

P(T) = 0.0914

P(T|C) = 0.85

Next, we need to calculate P(C|T=-22).

P(T=-22|C) = 1 - P(T>30 or T<-15|C)

P(T>30 or T<-15|C) = P(T>30|C) + P(T<-15|C) - P(T>30 and T<-15|C)

P(T>30|C) = 8/365

P(T<-15|C) = 16/365

P(T>30 and T<-15|C) = 0 (because the two events are mutually exclusive)

P(T>30 or T<-15|C) = 8/365 + 16/365 - 0 = 24/365

P(T=-22|C) = 1 - 24/365 = 341/365

P(T=-22) = P(T=-22|C') * P(C') + P(T=-22|C) * P(C)

P(T=-22) = 1/3 * (1 - 0.04) + 0

P(T=-22) = 0.3067

Finally, we can calculate P(C|T).

P(C|T=-22) = P(T=-22|C) * P(C) / P(T=-22)

P(C|T=-22) = (341/365) * 0.04 / 0.3067 ≈ 0

Therefore, the probability that the car will not start given the temperature being -22C is approximately 0, rounded to four decimal places.

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The probability that the car will not start given the temperature being −22C is 16.67 percent.

The car does not start 4% of the time each year, so there is a 96% chance of it starting.

There are 365 days in a year, so the likelihood of the car not starting is 0.04 * 365 = 14.6 days per year.

On these 14.6 days per year, the likelihood that the temperature is above 30°C or below -15°C is 85 percent. This suggests that out of the 14.6 days when the car does not start, roughly 12.41 of them (85 percent) are on days when the temperature is above 30°C or below -15°C. That leaves 2.19 days when the temperature is between -15°C and 30°C.

On these days, there is a 4% probability that the car will not start if the temperature is between -15°C and 30°C.

To calculate the probability that the car will not start given that the temperature is -22°C:

P(not starting | temperature=-22) = P(temperature=-22 | not starting) * P(not starting) / P(temperature=-22)

Plugging in the values:

P(not starting | temperature=-22) = 0.04 * (2.19 / 365) / 0.00242541

Simplifying the calculation:

P(not starting | temperature=-22) ≈ 0.1667 or 16.67 percent.

Rounding this figure to four decimal places, we get 0.1667 as the final solution.

Note: The result should be rounded to the appropriate number of decimal places based on the level of precision desired.

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The implicit equation for the plane passing through the points (5,1,5), (6,1,2), and (4,5,10) is:

-12x - 8y + 4z + 48 = 0

The implicit equation for the plane passing through the points (5,1,5), (6,1,2), and (4,5,10) is obtained by finding the normal vector to the plane.
To find the normal vector, we can use the cross product of two vectors formed by the given points. Let's choose the vectors formed by (5,1,5) and (6,1,2), and (5,1,5) and (4,5,10).
Vector 1: (6-5, 1-1, 2-5) = (1, 0, -3)
Vector 2: (4-5, 5-1, 10-5) = (-1, 4, 5)
Now, take the cross product of Vector 1 and Vector 2:
N = Vector 1 x Vector 2
  = (1, 0, -3) x (-1, 4, 5)
  = (-12, -8, 4)
The normal vector to the plane is (-12, -8, 4).
Now, using the equation of a plane in general form, Ax + By + Cz + D = 0, we can substitute the coordinates of any of the given points to find the value of D.
Using the point (5,1,5):
-12(5) - 8(1) + 4(5) + D = 0
-60 - 8 + 20 + D = 0
-48 + D = 0
D = 48

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The simplified expression is 26 - 4x.

To simplify the expression 6 - 4(x - 5), we can apply the distributive property and simplify the terms.

6 - 4(x - 5)

First, distribute -4 to the terms inside the parentheses:

6 - 4x + 20

Now, combine like terms:

(6 + 20) - 4x

Simplifying further:

26 - 4x

Therefore, the simplified expression is 26 - 4x.

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Answer:

page 10

Step-by-step explanation:

10+11+12+13+14+15=75

Hypothesis: An angle with a measure less than 90.

Conclusion: It is an acute angle.

The hypothesis of the conditional statement is "An angle with a measure less than 90," while the conclusion is "is an acute angle."

In a conditional statement, the hypothesis is the initial condition or the "if" part of the statement, and the conclusion is the result or the "then" part of the statement. In this case, the hypothesis states that the angle has a measure less than 90. The conclusion asserts that the angle is an acute angle.

An acute angle is defined as an angle that measures less than 90 degrees. Therefore, the conclusion aligns with the definition of an acute angle. If the measure of an angle is less than 90 degrees (hypothesis), then it can be categorized as an acute angle (conclusion).

Conditional statements are used in logic and mathematics to establish relationships between conditions and outcomes. The given conditional statement presents a hypothesis that an angle has a measure less than 90 degrees, and based on this hypothesis, the conclusion is drawn that the angle is an acute angle.

Understanding the components of a conditional statement, such as the hypothesis and conclusion, helps in analyzing logical relationships and drawing valid conclusions. In this case, the conclusion is in accordance with the definition of an acute angle, which further reinforces the validity of the conditional statement.

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(a) P(0) = 9000, P(4) ≈ 23051.

(b) The population will reach 18,000 in approximately 5 years.

(a). To find the population at time t=0, we substitute t=0 into the population growth function:

P(0) = 9000(1.3)[tex]^0[/tex] = 9000

To find the population at time t=4, we substitute t=4 into the population growth function:

P(4) = 9000(1.3)[tex]^4[/tex] ≈ 23051

Therefore, the population at time t=0 is 9000 and the population at time t=4 is approximately 23051.

(b). To determine when the population will reach 18,000, we need to solve the equation:

18000 = 9000(1.3)[tex]^t[/tex]

Divide both sides of the equation by 9000:

2 = (1.3)[tex]^t[/tex]

To solve for t, we can take the logarithm of both sides using any base. Let's use the natural logarithm (ln):

ln(2) = ln((1.3)[tex]^t[/tex])

Using the logarithmic property of exponents, we can bring the exponent t down:

ln(2) = t * ln(1.3)

Now, divide both sides of the equation by ln(1.3) to isolate t:

t = ln(2) / ln(1.3) ≈ 5.11

Therefore, the population will reach 18,000 in approximately 5 years.

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(a) The vectors 1, U2, ..., Uk in Rn are orthogonal if their dot products are zero for all pairs of distinct vectors. In other words, for i ≠ j, the dot product of Ui and Uj is zero: Ui · Uj = 0.

(b) The vectors v₁, U2, ..., Uk in Rn are orthonormal if they are orthogonal and have unit length. That is, each vector has a length of 1, and their dot products are zero for distinct vectors: ||v₁|| = ||U2|| = ... = ||Uk|| = 1, and v₁ · Uj = 0 for i ≠ j.

(c) To show that ATA is diagonal, we need to prove that the off-diagonal elements of ATA are zero. ATA = (A^T)(A), so the (i, j)-th entry of ATA is the dot product of the i-th column of A^T and the j-th column of A. If the columns of A are orthogonal, then the dot product is zero for i ≠ j, making the off-diagonal entries of ATA zero.

(d) If ATA is the identity matrix, it means that the dot product of the i-th column of A^T and the j-th column of A is 1 for i = j and 0 for i ≠ j. This implies that the columns of A form an orthonormal basis of Rn.

(e) The matrix P given in the question has columns that are unit vectors and orthogonal to each other. Therefore, the columns of P form an orthonormal basis of R³.

(f) The inverse of P can be found by taking the transpose of P since P is an orthogonal matrix. Therefore, the inverse of P is P^T.

(g) To solve the linear system of equations using P, we can use the equation X = PY, where X is the vector of unknowns and Y is the vector of knowns. Taking the inverse of P, we have X = P^T Y. By substituting the values of P and Y, we can calculate X.

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The  particular solution to the given differential equation is

[tex]$ \rm y_p(x) = \left(\frac{3}{5} - \frac{x}{5}\right) e^x$[/tex]

To find a particular solution to the given differential equation using the Method of Undetermined Coefficients, we assume a particular solution of the form:

[tex]\rm yp(x) = (A + Bx) e^x[/tex]

where A and B are constants to be determined.

Now, let's differentiate yp(x) with respect to x:

[tex]$ \rm y_p'(x) = (A + Bx) e^x + Be^x$[/tex]

[tex]$ \rm y_p''(x) = (A + 2B + Bx) e^x + 2Be^x$[/tex]

Substituting these derivatives into the differential equation, we have:

[tex]$ \rm (A + 2B + Bx) e^x + 2Be^x - 7[(A + Bx) e^x + Be^x] + 8(A + Bx) e^x = x e^x$[/tex]

Simplifying the equation, we get:

$(A + 2B - 7A + 8A) e^x + (B - 7B + 8B) x e^x + (2B - 7B) e^x = x e^x$

Simplifying further, we have:

[tex]$ \rm (10A - 6B) e^x + (2B - 7B) x e^x = x e^x$[/tex]

Now, we equate the coefficients of like terms on both sides of the equation:

[tex]$\rm 10A - 6B = 0\ \text{(coefficient of e}^x)}[/tex]

[tex]-5B = 1\ \text{(coefficient of x e}^x)[/tex]

Solving these two equations, we find:

[tex]$ \rm A = \frac{3}{5}$[/tex]

[tex]$B = -\frac{1}{5}$[/tex]

As a result, the specific solution to the given differential equation is:

[tex]$ \rm y_p(x) = \left(\frac{3}{5} - \frac{x}{5}\right) e^x$[/tex]

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The values of a and b that make the equation true are a = 4 and b = -45.

Let's simplify the equation first and then determine the values of a and b.

The given equation is: [tex]\[(4 + \sqrt{-49}) - 2(\sqrt{-4^2} + \sqrt{-324}) = a + bi\][/tex]

We notice that the terms inside the square roots result in complex numbers because they involve the square root of negative numbers. Therefore, we'll use complex numbers to simplify the equation.

[tex]\(\sqrt{-49} = \sqrt{49 \cdot -1} = \sqrt{49} \cdot \sqrt{-1} = 7i\)\(\sqrt{(-4)^2} = \sqrt{16 \cdot -1} = \sqrt{16} \cdot \sqrt{-1} = 4i\)\(\sqrt{-324} = \sqrt{324 \cdot -1} = \sqrt{324} \cdot \sqrt{-1} = 18i\)[/tex]

Now, substituting these values back into the equation:

(4 + 7i) - 2(4i + 18i) = a + bi

Simplifying further:

4 + 7i - 8i - 36i = a + bi

4 - i(1 + 8 + 36) = a + bi

4 - 45i = a + bi

Comparing the real and imaginary parts, we can determine the values of a and b:

a = 4

b = -45

Therefore, the values of a and b that make the equation true are a = 4 and b = -45.

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The answer is:5.56 seconds (rounded to the nearest 100th of a second).Given,The equation that describes the height of the rocket, y in feet, as it relates to the time after launch, x in seconds, is as follows: y = − 16x² + 89x+ 50.

To find the time that the rocket will hit the ground, we must set the height of the rocket, y to zero. Therefore:0 = − 16x² + 89x+ 50. Now we must solve for x. There are a number of ways to solve for x. One way is to use the quadratic formula: x = − b ± sqrt(b² − 4ac)/2a,

Where a, b, and c are coefficients in the quadratic equation, ax² + bx + c. In our equation, a = − 16, b = 89, and c = 50. Therefore:x = [ - 89 ± sqrt( 89² - 4 (- 16) (50))] / ( 2 (- 16))x = [ - 89 ± sqrt( 5041 + 3200)] / - 32x = [ - 89 ± sqrt( 8241)] / - 32x = [ - 89 ± 91] / - 32.

There are two solutions for x. One solution is: x = ( - 89 + 91 ) / - 32 = - 0.0625.

The other solution is:x = ( - 89 - 91 ) / - 32 = 5.5625.The time that the rocket will hit the ground is 5.5625 seconds (to the nearest 100th of a second). Therefore, the answer is:5.56 seconds (rounded to the nearest 100th of a second).

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The time that the rocket would hit the ground is 2.95 seconds.

Based on the information provided, we can logically deduce that the height (h) in feet, of this rocket above the​ ground is related to time by the following quadratic function:

h(t) = -16x² + 89x + 50

Generally speaking, the height of this rocket would be equal to zero (0) when it hits the ground. Therefore, we would equate the height function to zero (0) as follows:

0 = -16x² + 89x + 50

16t² - 89 - 50 = 0

[tex]t = \frac{-(-80)\; \pm \;\sqrt{(-80)^2 - 4(16)(-50)}}{2(16)}[/tex]

Time, t = (√139)/4

Time, t = 2.95 seconds.

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The given identity sinθtanθ = secθ - cosθ is not true. It does not hold for all values of θ.

To verify the given identity, we need to simplify both sides of the equation and check if they are equal for all values of θ.

Starting with the left-hand side (LHS), we have sinθtanθ. We can rewrite tanθ as sinθ/cosθ, so the LHS becomes sinθ(sinθ/cosθ). Simplifying further, we get sin²θ/cosθ.

Moving on to the right-hand side (RHS), we have secθ - cosθ. Since secθ is the reciprocal of cosθ, we can rewrite secθ as 1/cosθ. So the RHS becomes 1/cosθ - cosθ.

Now, if we compare the LHS (sin²θ/cosθ) and the RHS (1/cosθ - cosθ), we can see that they are not equivalent. The LHS involves the square of sinθ, while the RHS does not have any square terms. Therefore, the given identity sinθtanθ = secθ - cosθ is not true for all values of θ.

In conclusion, the given identity does not hold, and it is not a valid trigonometric identity.

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Abigail needs to pay $1,045.38 at the end of every 6 months to settle the loan in 5 years, and the amount of interest charged on the loan over the 5-year period is $0.00.

a) The amount to be paid at the end of every 6 months is $1,045.38. The loan is to be paid back in 5 years, which is 10 half-year periods. The principal amount borrowed is $34,550. The annual interest rate is 5.75%. The semi-annual rate can be calculated as follows:

i = r/2, where r is the annual interest rate

i = 5.75/2%

= 0.02875

P = 34550

PVIFA (i, n) = (1- (1+i)^-n) / i,

where n is the number of semi-annual periods

P = 34550

PVIFA (0.02875,10)

P = $204.63

The amount payable every half year can be calculated using the following formula:

R = (P*i) / (1- (1+i)^-n)

R = (204.63 * 0.02875) / (1- (1+0.02875)^-10)

R = $1,045.38

Hence, the amount to be paid at the end of every 6 months is $1,045.38.

b) The total amount paid by Abigail at the end of 5 years will be the sum of all the semi-annual payments made over the 5-year period.

Total payment = R * n

Total payment = $1,045.38 * 10

Total payment = $10,453.81

Interest paid = Total payment - Principal

Interest paid = $10,453.81 - $34,550

Interest paid = -$24,096.19

This negative value implies that Abigail paid less than the principal amount borrowed. This is because the interest rate on the loan is greater than the periodic payment made, and therefore, the principal balance keeps growing throughout the 5-year period. Hence, the interest charged on the loan over the 5-year period is $0.00 (rounded to the nearest cent).

Conclusion: Abigail needs to pay $1,045.38 at the end of every 6 months to settle the loan in 5 years, and the amount of interest charged on the loan over the 5-year period is $0.00.

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