Use completing the square to rewrite the given function in vertex form. Determine whether the vertex is a maximum or minimum
value. Select the TWO correct answers.
y = x2 + 4x - 10
-
A)
The vertex form is y = (x - 6)2 + 2.
B)
The vertex form is y= 4x + 2)2 - 10.
C)
The vertex form is y = -(x - 2)2 - 6.
D)
The vertex is a minimum value at point (-6,2).
E)
The vertex is a maximum value at point (2.-6).

Each year the account earns 5% interest, and you deposit 8% of your annual income with an annual income of $30000 which is growing at a continuous rate of 3% per year. The given differential equation models this situation dy/dt= 0.05y+2400e^(0.03*t)

First, let's break down the information given in the problem:
- y(t) represents your retirement account balance in dollars after t years.
- The account earns 5% interest each year, which means that the balance increases by 5% of the previous year's balance.
- You deposit 8% of your annual income into the account each year. Your current annual income is $30,000, but it is growing at a continuous rate of 3% per year.
Now, let's write the differential equation to model this situation. We know that the rate of change of y(t) is equal to the sum of the interest earned and the deposits made each year. In other words:
dy/dt = rate of interest + rate of deposits
The rate of interest is simply 5% of the current balance, or 0.05y. The rate of deposits is 8% of your current annual income, which is $30,000 plus 3% of your current annual income for each additional year. We can express this as \(0.08(30000)(1.03)^t\). Putting it all together, we get:
dy/dt = 0.05y + \(0.08(30000)(1.03)^t\)
But this isn't quite in the form of our desired answer yet. We can simplify the right-hand side by factoring out 0.05y, which gives us:
dy/dt = \(0.05y(1 + 1.6(1.03)^t)\)
Now we have our differential equation in the form of y' = ky, where k = \(0.05(1 + 1.6(1.03)^t)\). To solve this equation, we can use the separation of variables:
dy/y = k dt
Integrating both sides gives us:
ln|y| = kt + C
where C is the constant of integration. Exponentiating both sides gives us:
|y| = \(e^{(kt+C)}\) = \(Ce^{(kt)}\)
where C is a constant that depends on the initial conditions (i.e. the value of y when t=0). We can solve for C using the fact that y(0) = some initial value, say $10,000:
|10000| = \(Ce^{(0)}\)
C = 10000
So our final solution is:
y(t) = \(10000e^{(kt)}\)
where k is still equal to 0.05(1 + 1.6(1.03)^t). If we substitute this expression for k and simplify, we get:
y(t) = \(10000e^{(0.05t + 2400(e^{(0.03t)-1)/0.03)}}\)
which is equivalent to the answer given: dy/dt = 0.05y + \(2400e^{(0.03t)}\)

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Answer:

6

Step-by-step explanation:

Let x by the unknown number.

Given:

Therefore, the equation representing the given information is:

\(\boxed{x^2+85=(x-17)^2}\)

To solve for x, expand the right side:

\(\implies x^2+85=(x-17)^2\)

\(\implies x^2+85=(x-17)(x-17)\)

\(\implies x^2+85=x^2-34x+289\)

Subtract x² from both sides of the equation:

\(\implies 85=-34x+289\)

Add 34x to both sides:

\(\implies 34x+85=289\)

Subtract 85 from both sides:

\(\implies 34x=204\)

Divide both sides by 34:

\(\implies x=6\)

Therefore the unknown number is 6.

Answer:

alternate interior angle tq

Between the ages of 22 and 50, John would spend $102,200 on smoking, and between the ages of 22 and 70, he would spend $175,200 on smoking.


1. First, find the number of years John smokes between ages 22 and 50, and between ages 22 and 70:
  - For ages 22 to 50: 50 - 22 = 28 years
  - For ages 22 to 70: 70 - 22 = 48 years

2. Calculate the number of packs John smokes per year, given that he smokes 2 packs a day:
  - 2 packs/day × 365 days/year = 730 packs/year

3. Calculate the cost of smoking per year, given that each pack costs $5:
  - 730 packs/year × $5/pack = $3,650/year

4. Calculate the total cost of smoking for both age ranges:
  - For ages 22 to 50: 28 years × $3,650/year = $102,200
  - For ages 22 to 70: 48 years × $3,650/year = $175,200

If John continues to smoke at the same rate, he would spend $102,200 on smoking between the ages of 22 and 50, and $175,200 if he continues until the age of 70.

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Answer:

Pam is correct.

Step-by-step explanation:

Divide 15 by 3 = 5

5 x 4 = 20, the total of toys she has.

The integral of f(x,y) over the region 1 ≤ x²y² ≤ e⁴ is equal to 16e² ln(e²) - 16e² + 16.

To evaluate this integral, we need to first determine the limits of integration. Since we have the condition 1 ≤ x²y² ≤ e⁴, we can rewrite this as 1/x² ≤ y² ≤ e⁴/x², which gives us the limits for y. For x, we have the condition that x²y² ≤ e⁴, which can be rewritten as x² ≤ e⁴/y², giving us the limits for x.

Thus, the integral can be written as:

∫ ln(x²y²)/(x²y²) dy dx

Now, we can solve this integral by using techniques such as substitution or integration by parts. For simplicity, we can use the fact that ln(xy) = ln(x) + ln(y) and split the integral into two parts:

∫ ln(x) dy dx + ∫ ln(y) dy dx

Evaluating the first integral, we get:

∫ ln(x) [ln(e⁴/x²) - ln(1/x²)] dx

= 8∫(from 1 to e²) ln(x) dx

= 8[xln(x) - x] (from 1 to e²)

= 8e² ln(e²) - 8e² + 8

Evaluating the second integral, we get:

∫ ln(y) [ln(e⁴/x²) - ln(1/x²)] dx

= 8∫ ln(y) dx

= 8[e² ln(e²) - e² + 1]

Adding the two integrals together, we get:

8e² ln(e²) - 8e² + 8 + 8e² ln(e²) - 8e² + 8

= 16e² ln(e²) - 16e² + 16

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Answer:

x=-2, -6 > pick the first answer choice !!!!!

Step-by-step explanation:

Let's solve your equation step-by-step.

|x+4|=2

Solve Absolute Value.

|x+4|=2

We know either x+4=2or x+4=−2

x+4=2 (Possibility 1)

x+4−4=2−4 (Subtract 4 from both sides)

x=−2

x+4=−2 (Possibility 2)

x+4−4=−2−4 (Subtract 4 from both sides)

x=−6

Answer:

x=−2 or x=−6

Answered by the ONE & ONLY #QUEEN herself aka #DRIPPQUEENMO!!!

HOPE THIS HELPED!!! BAHAHAHAHAHA

If line m and n are parallel, the value of x is determined as 10.

The value of x is determined by applying the following rules of parallel lines as shown below.

The corresponding angles of the parallel lines are equal,

the corresponding angles are given as;

8x + 50 = 130

now solve for x, by making x the subject of the formula.

8x = 130 - 50

8x = 80

divide through by 8

8x / 8 = 80 / 8

x = 10

Thus, the value of x is function of the corresponding angles.

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By using the definition of continuity and exploiting the fact that f(0) = 1, we were able to prove the existence of an open interval (a, b) containing 0 such that for any real number r within this interval, the function value f(r) is greater than 0.

First, let's recall the definition of continuity at a point. A function f is continuous at a point c if, for any positive number ε, there exists a positive number δ such that whenever x is within δ of c, the value of f(x) will be within ε of f(c).

Now, since f is continuous at 0, we can say that for any positive ε, there exists a positive δ such that if |x - 0| < δ, then |f(x) - f(0)| < ε.

Since f(0) = 1, the above inequality simplifies to |f(x) - 1| < ε.

We want to find an open interval (a, b) containing 0 such that for any r within this interval, f(r) > 0. Let's consider ε = 1 as an arbitrary positive number.

From the definition of continuity at 0, we can find a positive δ such that if |x - 0| < δ, then |f(x) - 1| < 1. This implies -1 < f(x) - 1 < 1, which further simplifies to 0 < f(x) < 2.

Now, consider the interval (a, b) = (-δ, δ). Since δ is positive, it ensures that 0 is within this interval. Also, since f(x) is continuous on this interval, we can conclude that f(r) > 0 for all r within (-δ, δ).

To prove this, let's take any r within (-δ, δ). Since r is within this interval, we have -δ < r < δ, which implies |r - 0| < δ. By the definition of continuity at 0, we know that |f(r) - 1| < 1. Therefore, 0 < f(r) < 2, and we can conclude that f(r) > 0.

Hence, we have shown that there exists an open interval (a, b) containing 0 such that for all r within this interval, f(r) > 0.

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Answer:

14n^2-36n+16 is the answer

The possible values of this random variable range from 0 to 12, with 0 meaning none of the 12 parts in the sample do not conform and 12 meaning that all 12 parts in the sample do not conform.

The random variable in this scenario is the number of parts in a sample of 12 parts that do not conform to customer requirements. This random variable can take on any value from 0 to 12, where 0 would mean that none of the 12 parts in the sample do not conform and 12 would mean that all 12 parts in the sample do not conform. This range of values for the random variable is based on the fact that out of the 500 machined parts, only 10 do not conform to customer requirements. Therefore, the probability of each individual part in the sample not conforming is only 10/500 or 2%. This means that it is very unlikely that all 12 parts in the sample will not conform, but still possible.

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Answer:

A

Step-by-step explanation:

The thing that is represented is the area of the walkway along the length of the flower patch. Therefore, option A is the correct answer.

Given that, Sam created function A to represent the total area taken total length and width A= 4:2 + 36.1 + 72

From the information given, Sam created function A to represent the total area taken up by the flower patch and walkway by multiplying the functions modelling the new total length and width.

Therefore, option A is the correct answer.

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Answer:

Step-by-step explanation:

Given a graph showing two solid lines that cross in an X pattern with the left quadrant shaded, you want to know the system of equations that is being graphed.

The answer choices tell you that the boundary lines have equations ...

The boundary lines on the graph are solid, so they are included in the solution set. Both inequalities will include the "or equal to" case. (This eliminates choices B and C.)

The shading on the graph is to the left of each line, where x-values are less than those on the line. The shading is above the line with positive slope.

To tell where the shading is for a given inequality, look at the inequality symbol and a variable with a positive coefficient. (Coefficient and constant values do not matter for the purpose of determining shading.)

Knowing how the shading works, you can reject choice D:

That is, for this system of equations, the shading would be in the right quadrant of the X where the lines cross.

The system of inequalities you're looking for is the one shown in the attachment (choice A). These have shading to the left of the solid line in both cases:

Answer:

The answer is 1.62%

Step-by-step explanation:

A credit card issuer charges an APR of 19.66%

The billing cycle is 30 days

We have to find the periodic interest rate. The periodic rate is found by dividing the APR by the number of billing periods in the year.

The value here becomes:

= 1.615%≈1.62%

Hence, the answer is 1.62%

Hope this helps:)

Answer:

Its C!

Step-by-step explanation:

Hexagon 6 sides, pentagon 5 sides, square 4 sides, triangle 3 sides!

Hope this helps, and don't forget to follow!

The required answer of the given polygons series C. triangle. In other words, the next step in the series is triangle.

Given, the following shapes hexagon has 6 sides, a pentagon has 5 sides, a square has 4 sides,

A hexagon has 6 sides, a pentagon has 5 sides, a square has 4 sides, and a triangle has 3 sides. These are all examples of polygons, which are closed shapes formed by straight lines. The number of sides determines the name and characteristics of the polygon. Each polygon has its own unique properties and formulas for calculating its area and perimeter.

Therefore, the required answer of the given polygons series C. triangle. In other words, the next step in the series is triangle.

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540° is your answer!

Step-by-step explanation:

12−6x>24

subtract both side by 12

12-12-6x>24-12

-6x>12

divide both side by -6

x>-2

x<-2

The volume of the given prism is 72 cubic centimeters

A prism is a solid geometric figure whose two ends are similar, equal, and parallel rectilinear figures, and whose sides are parallelograms.

From the given figure, one cube represents 1 cubic cm

From the given figure

length =6cm

width = 3cm

Height = 4cm

Determine the volume

Volume = length * width * height

Substitute the given parameters into the formula

Volume = 6cm * 3cm * 4cm

Volume = 18 * 4

Volume = 72 cubic centimeters

Hence the volume of the given prism as shown in the diagram is 72 cubic centimeters

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Answer:

7.1

Step-by-step explanation:

the radius is half of that or 1.5

soooooooo

A=π*1.5*2

The value of the midpoint between (1, 2) and (11,12) is,

⇒ (6, 7)\

We have to given that;

To find the midpoint between (1,2) and (11,12).

Since, A pair of numbers which describe the exact position of a point on a cartesian plane by using the horizontal and vertical lines is called the coordinates.

Now, By definition of midpoint, we get;

The midpoint between (1,2) and (11,12) is,

⇒ (1 + 11)/2, (2 + 12) / 2

⇒ (12/2, 14/2)

⇒ (6, 7)

Thus, The value of the midpoint between (1, 2) and (11,12) is,

⇒ (6, 7)\

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It would be very surprising to observe a random sample of 30-time intervals between eruptions with a mean of 97 minutes, given the population mean and standard deviation.

Based on the information provided, we know that the mean time between eruptions of Old Faithful is 85 minutes and the standard deviation is 21 minutes. We are also given a sample of 30-time intervals between eruptions, and we want to determine whether it would be surprising to observe a sample mean of 97 minutes.

To answer this question, we need to calculate the standard error of the mean (SEM), which is given by the formula:

SEM = σ / sqrt(n)

where σ is the population standard deviation (21 minutes), and n is the sample size (30). Plugging in the values, we get:

SEM = 21 / sqrt(30) ≈ 3.84

Next, we calculate the z-score for a sample mean of 97 minutes, using the formula:

z = (x - μ) / SEM

where x is the sample mean (97 minutes), μ is the population mean (85 minutes), and SEM is the standard error of the mean (3.84 minutes). Plugging in the values, we get:

z = (97 - 85) / 3.84 ≈ 3.13

Finally, we can look up the probability of observing a z-score of 3.13 or more extreme in a standard normal distribution table (or using a calculator or software). The probability turns out to be very small, less than 0.01.

Therefore, Given the population mean and standard deviation, it would be very unexpected to find a random selection of 30-minute gaps between eruptions with a mean of 97 minutes. This suggests that the sample may not be representative of the population, or there may be some other factors affecting the intervals between eruptions.

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In an introductory engineering course with a total of 420 students, the number of boys can be calculated using the given ratio. There are approximately 270 boys in the course.

According to the given ratio of nine boys to every five girls, we can calculate the number of boys in the course.

To find the number of boys, we need to determine the proportion of boys in the total student population. The total number of students is 420, and the ratio of boys to girls is 9:5.

First, we calculate the total number of parts in the ratio: 9 + 5 = 14 parts.

To find the number of boys, we divide the total number of students by the total number of parts in the ratio and multiply it by the number of parts representing boys: (9/14) * 420 ≈ 270.

Therefore, there are approximately 270 boys in the introductory engineering course.

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Answer:

Rachel receives $30 and Brayden receives $15.

Step-by-step explanation:

If Rachel’s share is twice more, then you need to divide 45/3 to get 15. That means that Rachel’s share is $30 while Brayden’s is $15.

one real eigenvalue. find this eigenvalue, its multiplicity, and the dimension of the corresponding eigenspace.

What is eigenvalue?

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that, when the linear transformation is applied to it, changes at most by a scalar factor. The factor by which the eigenvector is scaled is known as the associated eigenvalue, frequently denoted by lambda.

a) -4

b) 1

c) 1

Step-by-step explanation:

a) The matrix A is given by:

\(A=\left[\begin{array}{ccc}-3 & 0 & 1 \\2 & -4 & 2 \\-3 & -2 & 1\end{array}\right]\\\)

where lambda are the eigenvalues and I is the identity matrix. By replacing you obtain:

\(A-\lambda I=\left[\begin{array}{ccc}-3-\lambda & 0 & 1 \\2 & -4-\lambda & 2 \\-3 & -2 & 1-\lambda\end{array}\right]\)

and by taking the determinant:\(\begin{aligned}& {[(-3-\lambda)(-4-\lambda)(1-\lambda)+(0)(2)(-3)+(2)(-2)(1)]-[(1)(-4-\lambda)(-3)+(0)(2)(1-} \\& \lambda)+(2)(-2)(-3-\lambda)]=0 \\& -\lambda^3-6 \lambda^2-12 \lambda-16=0\end{aligned}\)

and the roots of this polynomial is:

\(\begin{aligned}& \lambda_1=-4 \\& \lambda_2=-1+i \sqrt{3} \\& \lambda_3=-1-i \sqrt{3}\end{aligned}\)

hence, the real eigenvalue of the matrix A is -4.

b) The multiplicity of the eigenvalue is 1.

c) The dimension of the eigenspace is 1 (because the multiplicity determines the dimension of the eigenspace)

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Complete Quetion

The matrix A= (−3 0 1, 2 −4 2, −3 −2 1) has one real eigenvalue. Find this eigenvalue, its multiplicity, and the dimension of the corresponding eigenspace. The eigen value = has multiplicity = and the dimension of the corresponding eigenspace is:_______.

To maximize storage volume within the given constraints, you should buy 6 units of cabinet Y and no units of cabinet X.

Since the goal is to maximize storage volume, we need to consider the cost, floor space, and storage capacity of each cabinet option. Cabinet X costs $10 per unit and has a storage capacity of 8 cubic feet, while Cabinet Y costs $20 per unit and has a storage capacity of 12 cubic feet.

Considering the budget of $140, purchasing six units of Cabinet Y would cost $120. This leaves $20 unspent. On the other hand, buying 14 units of Cabinet X would also cost $140, utilizing the entire budget.

Next, we need to consider the floor space constraint. Cabinet X requires 6 square feet of floor space per unit, while Cabinet Y requires 8 square feet of floor space per unit. The total available floor space is 72 square feet. To maximize storage volume within this constraint, we should calculate how many units of each cabinet can fit.

With Cabinet X, 72 square feet can accommodate 12 units (6 square feet per unit * 12 units = 72 square feet). However, choosing Cabinet X would result in a storage volume of only 96 cubic feet (8 cubic feet per unit * 12 units).

With Cabinet Y, 72 square feet can accommodate 9 units (8 square feet per unit * 9 units = 72 square feet). By purchasing 6 units of Cabinet Y, we would have a total storage volume of 72 cubic feet (12 cubic feet per unit * 6 units).

Therefore, by buying 6 units of Cabinet Y and not purchasing any units of Cabinet X, we maximize the storage volume to 72 cubic feet, utilizing the entire floor space and staying within the given budget.

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Answer:

Present Value = \(X [\frac{1}{(1 + 0.12)^{1} } + \frac{1}{(1 + 0.12)^{2} } + \frac{1}{(1 + 0.12)^{3} } + \frac{1}{(1 + 0.12)^{4} } + \frac{1}{(1 + 0.12)^{5} } ]\)

Step-by-step explanation:

To find - If discount rate is 12%, the present value of Rs X received at the end of each year for the next five years is equal to .... ?

Solution -

We know that, formula for finding the Present vale is given by

Present value = Future value / (1 + r)ⁿ

where r is the rate of interest

and n is Number of periods

Now,

Here in the question, we have

r = 12% = 12/100 = 0.12

n = 5

Also, Given that, we have received Rs X at the end of each year

So,

Present Value = \(\frac{X}{(1 + 0.12)^{1} } + \frac{X}{(1 + 0.12)^{2} } + \frac{X}{(1 + 0.12)^{3} } + \frac{X}{(1 + 0.12)^{4} } + \frac{X}{(1 + 0.12)^{5} }\)

                        = \(X [\frac{1}{(1 + 0.12)^{1} } + \frac{1}{(1 + 0.12)^{2} } + \frac{1}{(1 + 0.12)^{3} } + \frac{1}{(1 + 0.12)^{4} } + \frac{1}{(1 + 0.12)^{5} } ]\)

⇒Present Value = \(X [\frac{1}{(1 + 0.12)^{1} } + \frac{1}{(1 + 0.12)^{2} } + \frac{1}{(1 + 0.12)^{3} } + \frac{1}{(1 + 0.12)^{4} } + \frac{1}{(1 + 0.12)^{5} } ]\)

Answer:

20 month

Step-by-step explanation:

The unknown here is the number of month required for plans A and B to be the same.
Let x be that month.

for both plans to be equal

19.25x + 200 = 26.75x + 50

19.25x-26.75x = 50-200

-7.5x = -150

x = 20

Answer:

there was a 44% increase to the number of bars.

Step-by-step explanation:

8/12 = 0.66

0.66 x 100 = 66

100 - 66 = 44

44%