what is 5 less than the square of a number in an algebraic expression

The red arc or chord in the key of C, or the solution to the provided question based on the circle, is 11.

A chord is a piece of a straight line that connects two points on a circle's circumference. When it crosses the circle at two different locations, it is also occasionally referred to as a secant.

The following formula can be used to determine a chord's length:

chord length = 2*radius*sin(angle/2)

where angle is the central angle that the chord is subtended by, and radius is the radius of the circle. In geometry and trigonometry, chords are frequently used to compute circle properties including area, circumference, and arc length.

Since the circle P ≅ circle C

In circle P the radius of PN =7 and

chord LM = 11 with an angle 104°

And  In circle C  the radius =7 and Circle and chord QR are both making the same angle. P = 104°

So the circle P ≅ circle C

The red arc or chord in C is consequently 11.

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The mean per capita income in thousands of dollars with 99% confidence interval and sample size of 1417 is equal to CI = (23.7, 24.3).

Construct the 99% confidence interval for the mean per capita income, use the formula,

CI = x ± Z× (σ / √n)

where

x is the sample mean,

σ is the population standard deviation,

n is the sample size,

Z is the z-score corresponding to the desired level of confidence.

Using attached z-score table,

For a 99% confidence interval, the corresponding z-score is 2.58

Substituting the given values, we get,

⇒CI = 24 ± 2.58 × (5.1 / √1417)

Simplifying the expression inside the parentheses, we get,

⇒CI = 24 ± 0.349

⇒CI = (23.7, 24.3)

Rounding to one decimal place, the confidence interval is (23.7, 24.3) thousands of dollars.

Therefore, the 99% confidence interval for the mean per capita income is CI = (23.7, 24.3).

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a. The expected proportion of significant errors per year at this hospital is 0.2%. b. We can expect the proportion of significant errors per year at this hospital to fall within the range of 0.15% to 0.25%.

a. The expected proportion of errors per year at this hospital can be calculated as follows.

Number of medications prescribed and dispensed per year = 60,000

Proportion of medications in error = 1/100 = 0.01

Expected number of medications in error per year = 60,000 x 0.01 = 600

Therefore, the expected proportion of errors per year at this hospital is 600/60,000 = 0.01 or 1%.

To calculate the expected proportion of significant errors per year, we need to know the proportion of errors that result in significant problems for the patient. From the given information, we know that 1 in 500 errors resulted in significant problems. Therefore, the proportion of significant errors is 1/500 = 0.002.

Expected number of significant errors per year = 60,000 x 0.002 = 120

Therefore, the expected proportion of significant errors per year at this hospital is 120/60,000 = 0.002 or 0.2%.

b. To calculate the 2-σ interval for the proportion of significant errors per year, we need to use the formula:

2-σ interval = expected proportion ± 2 x standard error

The standard error can be calculated as follows:

Standard error = sqrt(p(1-p)/n)

where p is the expected proportion of significant errors (0.002) and n is the number of medications prescribed and dispensed per year (60,000)

Standard error = sqrt(0.002 x 0.998/60,000) = 0.000246

Substituting the values in the formula, we get:

2-σ interval = 0.002 ± 2 x 0.000246

2-σ interval = 0.001509 to 0.002491 (rounded to 5 decimal places)

Therefore, we can expect the proportion of significant errors per year fall within the range of 0.001509 to 0.002491 or 0.15% to 0.25%.

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Answer:

AB = 15

Step-by-step explanation:

6(6 + x + 6) = 7(7 + 11)

72 + 6x = 126

6x = 126 - 72 = 54

x = 54/6

   = 9.

So AB = 9 + 6 = 15.

Area of sector = 161.1 square inches

Activity 1:

The radius of the pizza is half of its diameter, which is 16/2 = 8 inches.

The central angle of the sector is 80%, which is 0.8 times 360 degrees = 288 degrees.

To find the area of the sector, we use the formula:

Area of sector = (central angle / 360) x πr^2

Area of sector = (288 / 360) x π x 8^2

Area of sector = (0.8) x π x 64

Area of sector = 161.1 square inches (rounded to the nearest tenth)

Activity 2:

The unit of measurement for the area of the sector is square inches (in²).

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Answer:

To make the polynomial 9x^10 + ax^b + 100 a perfect square trinomial, we need to add a constant term to it such that it becomes a square of a binomial.

Let's first write the square of a binomial in general form:

(a + b)^2 = a^2 + 2ab + b^2

If we compare this general form with our polynomial, we can see that the first term, 9x^10, is equal to (3x^5)^2, which means that we can write our polynomial as:

(3x^5)^2 + ax^b + 100 = (3x^5 + c)^2

Expanding the right-hand side of this equation, we get:

(3x^5 + c)^2 = 9x^10 + 6cx^15 + c^2

Comparing the coefficient of x^15 on both sides, we get:

6c = 0

Since c cannot be zero (otherwise we would end up with the original polynomial), this means that we must have:

c = 0

Therefore, we can write our polynomial as:

(3x^5)^2 + ax^b + 100 = (3x^5)^2

Expanding the right-hand side, we get:

(3x^5)^2 = 9x^10

Therefore, we must have:

a = 0

b = 10

So the values of a and b that make the polynomial 9x^10 + ax^b + 100 a perfect square trinomial are a = 0 and b = 10.

The approximate total amount of string needed to keep the tree perpendicular to the ground is 4.02 feet, which is closest to answer choice C, 5.83 ft.

Assuming that Luis attached the ropes at the same height on the tree, the length of the rope needed for each side of the tree would be equal to the distance from the tree to the stake.

To keep the tree perpendicular to the ground, the distance from the tree to the stake should be equal to half of the diameter of the tree's canopy.

However, since the diameter of the canopy is not given, we can estimate it based on the height of the tree.

According to some tree experts, the average height-to-canopy-diameter ratio for a mature tree is about 5:1.

This means that if the tree is 20 feet tall, its canopy diameter is approximately 4 feet.

Using this estimate, we can assume that the canopy diameter of Luis's tree is about 4 feet, or 1.33 yards.

Thus, the distance from the tree to the stake should be approximately 0.67 yards.

Since there are two sides of the tree, Luis would need a total of 2 times 0.67 yards, or approximately 1.34 yards of rope.

Converting yards to feet, we get:

[tex]1.34 yards * 3 feet/yard = 4.02 feet[/tex]

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Answer:

if vacuole are absent in plant cell then there is no storage of food and ions in the process of and permeability of cell may be distorted

1. The next two term for the sequence using Geometric Progression is 8 and 16

2. The next two terms for the sequence using arithmetic progression is 7 and 11

A sequence is an ordered list of numbers (or other elements like geometric objects), that often follow a specific pattern or function.

Using Geometric Progression, the common ratio is 2/1 = 2

therefore the next two terms will be

4× 2 = 8 and 8× 2 = 16

Using Arithmetic progression , the common difference will be increasing by 1 per number of term, i.e r+1

for the fourth term ,common difference = 2+1 = 3

fourth term = 4+3 = 7

for the fifth term , common difference = 3+1 = 4

fifth term = 7+4 = 11

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The vertical displacement of the mule comes out to be the difference between the final and the initial position which is 1190 m.

The displacement refers to the distance between the final and the initial position of an object. It is the shortest distance between these points is the displacement of the object. It is a vector quantity.

Vector quantity refers to the measurement in which both magnitude and direction are considered.

Starting point = 1290 m

Final point = 2480 m

Displacement = 2480 - 1920

= 1190 m

1190 m is the vertical displacement of the mule when traveling from one post to another.

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The question is in Spanish and when translated to English, it is:

On a mule trip to Duarte Peak, the rider observes a post 1,290 m above sea level, after 5 hours of walking he pays attention to another post that indicates 2,480 m above sea level. What has been its displacement in the vertical direction?

It is not possible to use n = 10.

It is not possible to use n = 20.

It is possible to use n = 25.

1. The error probabilities for n = 10 are as follows:

- P-value: It is not possible to use n = 10 because there is no value of x which results in a p-value.
- Beta (0.3): B(0.3) > 0.1, which means the probability of failing to reject the null hypothesis (p = 0.1) when the true probability of a leak is actually 0.3 is too high.

Therefore, it is not possible to use n = 10.

2. The error probabilities for n = 20 are as follows:

- P-value: It is not possible to use n = 20 because it results in a beta error probability (B(0.3)) < 0.1, which means the probability of incorrectly rejecting the null hypothesis (p = 0.1) when the true probability of a leak is actually 0.3 is too low.
- Beta (0.3): B(0.3) > 0.1, which means the probability of failing to reject the null hypothesis (p = 0.1) when the true probability of a leak is actually 0.3 is too high.

Therefore, it is not possible to use n = 20.

3. The error probabilities for n = 25 are as follows:

- P-value: P(0.3) < 0.1, which means the probability of incorrectly rejecting the null hypothesis (p = 0.1) when the true probability of a leak is actually 0.3 is low enough to proceed with production.
- Beta (0.3): B(0.3) < 0.1, which means the probability of failing to reject the null hypothesis (p = 0.1) when the true probability of a leak is actually 0.3 is low enough to proceed with production.

Therefore, it is possible to use n = 25.

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The only possible value of m that would make the ranges of f(x) and g(x) the same is any positive real number.

The range of a function is the set of all possible output values. In this case, we are given that the ranges of two functions, f(x) and g(x), are the same.

The function f(x) = √(mx) has a domain of x ≥ 0, since the square root of a negative number is not a real number. The function g(x) = m√x has a domain of x ≥ 0 for the same reason.

To find the range of these functions, we need to consider the possible values of the input x. For f(x), as x increases, the output √(mx) also increases, and as x approaches infinity, so does the output. For g(x), as x increases, the output m√x also increases, and as x approaches infinity, so does the output.

Therefore, if the ranges of f(x) and g(x) are the same, this means that they both have the same maximum and minimum values, and these values are achieved at the same inputs.

In particular, if we consider the minimum value of the range, this is achieved when x = 0, since both functions are defined only for non-negative inputs. At x = 0, we have f(0) = g(0) = 0, so the minimum value of the range is 0.

To find the maximum value of the range, we need to consider the behavior of the functions as x approaches infinity. As noted above, both functions increase without bound as x increases, so the maximum value of the range is infinity.

Therefore, the only possible value of m that would make the ranges of f(x) and g(x) the same is any positive real number.

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The unit vector in the direction of maximum increase of f(x,y,z) at P is:

v = (∇f(-1,-1,-1)) / ||∇f(-1,-1,-1)|| = (2/2√3, 2/2√3, 2/2√3) = (√3/3, √3/3, √3/3)

a. The gradient of f(x,y,z) is given by the vector ∇f(x,y,z) = (∂f/∂x, ∂f/∂y, ∂f/∂z). Using the partial derivative rules, we have:

∂f/∂x = 2yz

∂f/∂y = 2xz

∂f/∂z = 2xy

Therefore, the gradient of f(x,y,z) is:

∇f(x,y,z) = (2yz, 2xz, 2xy)

Evaluating this at P(-1,-1,-1), we get:

∇f(-1,-1,-1) = (2(-1)(-1), 2(-1)(-1), 2(-1)(-1)) = (2,2,2)

b. The unit vector in the direction of maximum increase of f(x,y,z) at P is given by the unit vector in the direction of ∇f(-1,-1,-1). Since ∇f(-1,-1,-1) = (2,2,2), the unit vector in the direction of ∇f(-1,-1,-1) is:

v = (∇f(-1,-1,-1)) / ||∇f(-1,-1,-1)||

where ||∇f(-1,-1,-1)|| is the magnitude of the gradient vector, which is:

||∇f(-1,-1,-1)|| = sqrt((2)^2 + (2)^2 + (2)^2) = 2√3

Therefore, the unit vector in the direction of maximum increase of f(x,y,z) at P is:

v = (∇f(-1,-1,-1)) / ||∇f(-1,-1,-1)|| = (2/2√3, 2/2√3, 2/2√3) = (√3/3, √3/3, √3/3)

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The inequality which is used to represent normal "temperature-range" for  "human-body", is (d) |t − 98.6| ≤ 1.8.

The "average-temperature" of body is = 98.6° F, and it can vary by 1.8°F.

The inequality |t − 98.6| ≤ 1.8 indicates that the absolute difference between the body temperature and the average temperature is less than or equal to 1.8° F.

This means that the body temperature t can vary within a range of 1.8° F from the average temperature of 98.6° F.

Which means, the temperature cam range from :

⇒ 98.6-1.8 ≤ t ≤ 98.6+1.8,

⇒ 96.8 ≤ t ≤ 100.4;

Therefore, the correct inequality is (d) |t − 98.6| ≤ 1.8.

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The given question is incomplete, the complete question is

The average human body temperature is 98.6° F, but it can vary by as much as 1.8° F. Write an inequality to represent the normal temperature range of the human body, where t represents body temperature.

(a) |t − 1.8| ≥ 98.6

(b) |t − 1.8| ≤ 98.6

(c) |t − 98.6| ≥ 1.8

(d) |t − 98.6| ≤ 1.8

Answer: |t − 98.6| ≤ 1.8

Step-by-step explanation: If takes then takes then takes then takes then takes.

Answer:

5.6 miles

Step-by-step explanation:

the patteren in 0.3    0.6     0.3       0.6

Mike will need 28 feet of fencing.

To solve the problem, we can use the formula for the area of a rectangle:

A = L × W

where A is the area, L is the length, and W is the width.

We know that the area of the patio is 192 ft^2, so we can write:

192 = L × W

We also know that the length is 4 feet longer than the width, so we can write:

L = W + 4

Substituting L = W + 4 into the equation for the area, we get:

192 = (W + 4) × W

Expanding the right side of the equation, we get:

192 = W^2 + 4W

Rearranging, we get a quadratic equation in standard form:

W^2 + 4W - 192 = 0

We can solve for W by factoring or using the quadratic formula, but in this case, we can recognize that 12 and -16 are two numbers that multiply to -192 and add up to 4. Therefore, we can write:

W^2 + 4W - 192 = (W + 16) × (W - 12) = 0

This gives us two possible values for W: W = -16 or W = 12. Since the width cannot be negative, we reject the solution W = -16 and choose W = 12.

Using the equation L = W + 4, we find that the length is L = 16.

Finally, we can calculate the amount of fencing Mike will need by adding up the lengths of the three sides that need to be fenced. The two lengths are L = 16 feet each, and the width is W = 12 feet. Therefore, Mike will need a total of 16 + 16 + 12 = 44 feet of fencing. However, since one side of the patio is adjacent to the back of his house, he only needs to fence three sides.

Therefore, he will need 44 - 16 = 28 feet of fencing.

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The volume of the sand pile to the nearest cubic meter would be 10,121 cubic meters.

To find the volume of the sand pile, we need to know its base dimensions and height. Since we have the angle and the length of the slant side (RT) of the pile, we can use trigonometry to find the height and base dimensions.

We can use the sine function to find the height (TO):

sin(R) = opposite / hypotenuse

sin(40) = TO / 20

We can also use the cosine function to find the radius (RO):

cos(R) = adjacent / hypotenuse

cos(40) = RO / 20

Calculate the values:

TO = 20 x sin(40) = 12.85 meters

RO = 20 x cos(40) = 15.32 meters

Finally, we can find the volume V of the cone-shaped sand pile using the formula:

V = (1/3) x π x r² x h

V = (1/3) x π x (15.32)² x 12.85

V = 10,121.39 cubic meters

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The rule for the reflection shown above is (x, y) → (x, -y).

In Mathematics and Geometry, a reflection over or across the x-axis is represented by this transformation rule (x, y) → (x, -y).

This ultimately implies that, a reflection over or across the x-axis would maintain the same x-coordinate while the sign of the y-coordinate changes from positive to negative or negative to positive.

Conversely, a reflection over or across the y-axis would maintain the same y-coordinate while the sign of the x-coordinate changes from positive to negative or negative to positive.

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The angle between their courses is 42.02°.

It is important to first find the distance between them after the 2 hours of travel.

Recall the  formula:

speed = distance/time

Make distance the subject of the formula

distance = speed x time

For the oil tanker,

given the following:

speed = 32mph

time = 2hr

distance = 32 mph x 2 hours = 64 miles

For the cruise ship,

given the following:

speed = 46 mph,

time = 2 hr

distance = 46 mph x 2 hours = 92 miles

So after two hours of travel, the two vessels are 63 miles apart. This means that they are forming a triangle with the distance between them as the longest.

Now we need to find the angle between the two vessels' courses by using the Cosine rule:

Recall that

a² = b² + c² -2bc Cos A

Let C be the angle between the oil tanker and cruise ship

then we can rewrite the equation as:

c² = a² + b² -2bc Cos C

where

a = 64miles (distance of oil tanker)

b = 92miles (dsitance of cruise ship)

c = 63miles (distance between the vessels)

C = angle between the vessels

Plug in the values to the equation

63² = 64² + 92² - 2(64)(92) Cos C

3969 = 4096 + 8464 - 11776 Cos C

3969 = 12560 - 11776 Cos C

Collect like terms

3969 - 12560 = - 11776 Cos C

8591 = 11776 Cos C

Cos C = 8591/11776

Cos C = 0.7295

Apply the inverse Cosine formula

C = Cos⁻¹ (0.7295)

C = 42.02°

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In order for a triangle to be acute, the relationship that c² must have with a² + b² is c² < a² + b².

An acute triangle is a triangle in which all three angles are acute angles, which means they are less than 90 degrees. In other words, an acute triangle is a triangle with three acute angles.

To understand why the relationship between c^2 (the square of the longest side) and a^2 + b^2 (the sum of the squares of the other two sides) is important in determining whether a triangle is acute, we need to delve into the concept of the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, it can be expressed as c^2 = a^2 + b^2, where c represents the hypotenuse, and a and b represent the other two sides.

In an acute triangle, the sum of the squares of the two shorter sides must be greater than the square of the longest side.

This can be visualized as follows: If we were to draw a right triangle with the shorter sides represented by segments a and b, and the longest side represented by segment c, the acute triangle would be formed by making the length of segment c shorter than the length determined by the Pythagorean theorem. This ensures that the angle opposite to the longest side remains acute.

On the other hand, if c^2 were equal to a^2 + b^2, we would have a right triangle, not an acute triangle. If c^2 were greater than a^2 + b^2, we would have an obtuse triangle since the angle opposite to the longest side would be greater than 90 degrees.

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Answer:

x = 73, y = 88, z = 45

Step-by-step explanation:

40+52+y = 180 (Angle Sum Property)

=> y = 180-40-52

=> y = 88

x + (15 + 40) + 52 = 180 (Angle Sum Property)

=>x = 180 - 52 - 55

=> x = 73

40 + 43 + (52+z)  = 180

=> z = 180 -53 - 40 -43

=> z = 45

The measure of center that best represents the data is mean and its values for José and Manuel are 20.1 and 21.8, respectively. Comparing the mean values, José generally scores less goals in a game than Manuel.

To find the measure of center that best represents the data, we will use the mean.

The measure is calculated by adding up all the values and dividing by the total number of values.

The mean number of goals for José is:

= (7+12+17+4+18+25+38+32+37+11)/10

= 20.1

The mean number of goals for Manuel is:

= (17+23+21+31+30+5+26+37+19+9)/10

= 21.8.

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Margo is right; Jules made a 45 degree angle.

To determine who is right about the angle made by Jules while taking the first piece of pizza, we need to compare their estimates of 20 degrees and 45 degrees.

Since angles are measured using a protractor or other measuring tools, we rely on accurate measurement techniques to determine their values. If both Jules and Margo used appropriate measuring tools and techniques, we would expect their measurements to be close.

However, a 20-degree angle is significantly smaller than a 45-degree angle. Therefore, based on the provided information, Margo's estimate of a 45-degree angle seems more reasonable and likely to be correct.

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Step-by-step explanation:

Los 10 m representan 1/3 del camino, ya que si sumamos 10 + 20 + 30, obtenemos la distancia total del camino, que es 60 m.

Si la casa se encuentra a 25 m del camino, entonces está a una distancia de 5 m del final del camino, ya que 25 + 5 = 30. Por lo tanto, la casa está a 2/3 del camino, es decir, a una fracción de 2/3 de la distancia total del camino.

La casa está representada a 2/3 del camino, lo que corresponde a una distancia de 40 m (2/3 de 60 m). Por lo tanto, la casa está representada a 40 m del comienzo del camino.

Los 20 m representan 1/3 del camino, ya que si sumamos 10 + 20 + 30, obtenemos la distancia total del camino, que es 60 m. Por lo tanto, los 20 m representan la misma fracción que los 10 m, que es 1/3 del camino.

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he estimated proportion of teens who smoked cigarettes in the last 30 days is 0.05815 or 5.815%. This result suggests that cigarette use among teens is down, as stated in the National Youth Tobacco Survey conducted by the US Centers for Disease Control.

It is mentioned that the 2019 preliminary results show that e-cigarette use is up among US teens while cigarette use is down. In the sample of 1582 teens, 92 reported smoking a cigarette in the last 30 days.

To estimate the proportion of teens who smoked cigarettes in the last 30 days, follow these steps:

Step 1: Find the total number of teens in the sample.
There were 1582 teens in the sample.

Step 2: Find the number of teens who reported smoking a cigarette in the last 30 days.
92 teens reported smoking a cigarette in the last 30 days.

Step 3: Calculate the proportion of teens who smoked cigarettes in the last 30 days.
Divide the number of teens who smoked cigarettes (92) by the total number of teens in the sample (1582).

Proportion = 92 / 1582 = 0.05815 (rounded to 5 decimal places)

So, the estimated proportion of teens who smoked cigarettes in the last 30 days is 0.05815 or 5.815%. This result suggests that cigarette use among teens is down, as stated in the National Youth Tobacco Survey conducted by the US Centers for Disease Control.

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After evaluating the conclusion is that one serving of orange juice has 35.7 mg more vitamin C per serving than one serving of cranberry  juice.

According to the provided data , a bottle of orange juice has 750 mg of vitamin C and provides 6 servings. A bottle of cranberry juice has 134 mg of vitamin C and provides 1.5 servings.

Now to evaluate how many milligrams of vitamin C are in 1 serving of each type of juice, we have to  perform division to evaluate the total amount of vitamin C by the number of servings.

For orange juice

750 mg / 6 servings

= 125 mg/serving

For cranberry juice

134 mg / 1.5 servings

= 89.3 mg/serving

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Keegan will have approximately $878.85 in the account after 5 years.

What is Compound interest ?

Compound interest is the interest that is earned not only on the initial amount of money invested (known as the principal), but also on any interest earned on that principal over time. In other words, compound interest is interest on interest.

The compound interest formula is given by:

A = P[tex](1 + r/n)^{nt}[/tex]

where A is the amount after t years, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

In this case, Keegan deposited $675, the annual interest rate is 4.8%, the interest is compounded quarterly, and the investment is for 5 years. Therefore, we can plug in these values into the formula to get:

A = 675[tex](1 + 0.048/4)^{20}[/tex]

A = 675[tex](1.012)^{20}[/tex]

A ≈ $878.85

Therefore, Keegan will have approximately $878.85 in the account after 5 years.

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The solution to the system of differential equations x' = y - x + t and y' = y with initial conditions x(0) = 9 and y(0) = 4 is x(t) = 10e^t - t - 1 and y(t) = 9e^t - 5t - 5.To find this solution, we first solve for y in the second equation:y' - y = 0y(t) = Ce^tNext, we substitute this expression for y into the first equation and solve for x:x' = Ce^t - x + tx' + x = Ce^t + tMultiplying both sides by e^t, we get:(e^t x)' = Ce^2t + te^tIntegrating both sides:e^t x(t) = (C/2)e^2t + te^t + DUsing the initial condition x(0) = 9, we get:D = 9Using the expression for y(t) and the initial condition y(0) = 4, we get:C = 5Substituting these values into the equation for x(t), we get:x(t) = 10e^t - t - 1Finally, we substitute the expression for y(t) into the given initial condition y(0) = 4 and solve for the constant C:C = 9 - 5tSubstituting this expression for C into the equation for y(t), we get:y(t) = 9e^t - 5t - 5

For more similar questions on topic Vectors in 2D is a sub-topic in linear algebra that deals with the study of vectors in two-dimensional space. In two-dimensional space, vectors are represented as ordered pairs of real numbers and can be used to describe quantities such as displacement, velocity, and force. The magnitude and direction of a vector can be calculated using trigonometry, and vectors can be added, subtracted, and multiplied by scalars using the rules of vector algebra.

In the context of the given problem, we are asked to find two unit vectors in 2D that make an angle of 45 degrees with a given vector 6i + 5j, where i and j are the unit vectors in the x and y directions, respectively. To solve this problem, we need to use the properties of vectors and trigonometry to find the appropriate unit vectors that satisfy the given conditions. The solution to this problem involves finding the components of the given vector, calculating the angle between this vector and the x-axis, and using this angle to construct the desired unit vectors.

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The solution to the system of differential equations is:
x(t) = 5 e^(t/2) - 4 e^(3t/2)
y(t) = 4

To solve this system of differential equations, we can use Laplace transforms. Taking the Laplace transform of both sides of each equation, we get:

sX(s) - x(0) = Y(s) - X(s) + T Y(s)
sY(s) - y(0) = Y(s)

Substituting in the initial conditions x(0) = 9 and y(0) = 4, we can solve for X(s) and Y(s):

X(s) = (s + 1)/(s^2 - s - T)
Y(s) = 4/s

To find x(t) and y(t), we need to inverse Laplace transform these expressions. We can use partial fractions to simplify the expression for X(s):

X(s) = A/(s - r1) + B/(s - r2)

where r1 and r2 are the roots of the denominator s^2 - s - T, given by:

r1 = (1 - sqrt(1 + 4T))/2
r2 = (1 + sqrt(1 + 4T))/2

Solving for A and B, we get:

A = (r2 + 1)/(r2 - r1)
B = -(r1 + 1)/(r2 - r1)

Substituting these values back into the expression for X(s), we get:

X(s) = (r2 + 1)/(r2 - r1)/(s - r1) - (r1 + 1)/(r2 - r1)/(s - r2)

Taking the inverse Laplace transform of this expression, we get:

x(t) = (r2 + 1)/(r2 - r1) e^(r1 t) - (r1 + 1)/(r2 - r1) e^(r2 t)

Substituting in the values for r1 and r2, we get:

x(t) = 5 e^(t/2) - 4 e^(3t/2)

Similarly, taking the inverse Laplace transform of Y(s) = 4/s, we get:

y(t) = 4

Therefore, the solution to the system of differential equations is:

x(t) = 5 e^(t/2) - 4 e^(3t/2)
y(t) = 4

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