Note: Figure is not drawn to scale. If the route takes him 10 miles on Forrest Lane and 26 miles up Cedar Drive, how far will Anthony ride down Pine Avenue?

x = 0, π, 2π, 3π, 4π, 5π, 6π, π/3, 5π/3

These are the exact solutions of the given equation on the interval [0, 21). To find all exact solutions of the equation sec(x) sin(x) - 2 sin(x) = 0 on the interval [0, 21), we will use the factoring method:

First, we can factor out the sin(x) term:

sin(x) (sec(x) - 2) = 0

Now, we have two separate equations to solve:

1) sin(x) = 0
2) sec(x) - 2 = 0

For equation (1), sin(x) = 0 at x = nπ, where n is an integer. We need to find the values of n that give solutions in the range [0, 21):

0 ≤ nπ < 21
0 ≤ n < 21/π
n = 0, 1, 2, 3, 4, 5, 6

x = 0, π, 2π, 3π, 4π, 5π, 6π

For equation (2), sec(x) - 2 = 0, or sec(x) = 2. We know that sec(x) = 1/cos(x), so:

1/cos(x) = 2
cos(x) = 1/2

The values of x for which cos(x) = 1/2 in the range [0, 21) are x = π/3 and x = 5π/3.

Combining both sets of solutions, we have:

x = 0, π, 2π, 3π, 4π, 5π, 6π, π/3, 5π/3

These are the exact solutions of the given equation on the interval [0, 21).

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The angle G from triangle FGH has a measure of approximately 1°.

In this problem we find the case of a triangle with two known sides and a known angle. By Euclidean geometry, the sum of all internal angles in a triangle equals 180° and we are required to find all possible values of angle G. This can be done by using sine law:

(980 in) / sin 170° = (140 in) / sin G

sin G = 0.024

G = 1.421°

The only possible value for angle G is equal to 1.421°.

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Answer:

k = - 0.36 or k = 30.36

Step-by-step explanation:

k² - 30k = 11

to complete the square

add ( half the coefficient of the k- term )² to both sides

k² + 2(- 15)k + 225 = 11 + 225

(k - 15)² = 236 ( take square root of both sides )

k - 15 = ± [tex]\sqrt{236}[/tex] ≈ ± 15.36 ( to the nearest hundredth )

add 15 to both sides

k = 15 ± 15.36

Then

k = 15 - 15.36 = - 0.36

or

k = 15 + 15.36 = 30.36

Answer:

push-up = 1 second

sit-up = 3 seconds

Step-by-step explanation:

let p represent the # of push-ups

let s represent the # of sit-ups

System of equations:

19p+8s=43

12p+12s=48

i'll eliminate s by multiplying the top equation by 3 and the bottom equation by -2

57p+24s=129

-24p-24s=-96

33p=33

p=1 second

now solve for s (i'll plug p into the 2nd equation)

12(1) + 12s=48

12s=36

s=3 seconds

To determine if x and y are proportional, we need specific values or a proportional relationship equation.

To determine whether x and y are proportional, we need to compare the ratio of their values. If the ratio of x to y remains constant as x and y vary, then they are proportional.

Mathematically, if x/y = k, where k is a constant, then x and y are proportional. However, without specific values or equations, it is not possible to ascertain their proportionality.

Without further information, we cannot determine whether x and y are proportional. Additional context, such as specific values or an equation relating x and y, is needed to make a conclusive statement about their proportionality.

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Answer: the minimum distance from P to the circle is approximately 2.97, and the maximum distance is approximately 3.89.

Step-by-step explanation:

To find the minimum and maximum distances from the point P(-3, 9) to the circle defined by (x-3)^2 + (y-1)^2 = 25, we can use the fact that these distances are given by the perpendiculars from the point P to the line passing through the center of the circle.

The center of the circle is (3,1), so we can find the equation of the line passing through P and the center of the circle as follows:

The slope of the line passing through P and the center of the circle is (1-9)/(3-(-3)) = -8/6 = -4/3.

Using the point-slope form of a line, the equation of the line passing through P and the center of the circle is y - 9 = (-4/3)(x + 3).

Now we can find the points where this line intersects the circle. Substituting y = (-4/3)(x+3) + 9 into the equation of the circle, we get:

(x-3)^2 + ((-4/3)(x+3) + 8)^2 = 25

Expanding and simplifying this equation gives a quadratic equation in x:

25x^2 + 96x + 80 = 0

Solving this quadratic equation using the quadratic formula, we get:

x = (-96 ± sqrt(96^2 - 42580)) / (2*25)

x = (-96 ± 56) / 50

x = -2.04 or x = -1.52

Substituting these values of x into y = (-4/3)(x+3) + 9 gives the corresponding values of y:

When x = -2.04, y = 6.24

When x = -1.52, y = 7.27

So the two points of intersection are approximately (-2.04, 6.24) and (-1.52, 7.27).

Finally, we can find the distances from P to each of these points using the distance formula:

The distance from P to (-2.04, 6.24) is sqrt[(-3 - (-2.04))^2 + (9 - 6.24)^2] ≈ 3.89.

The distance from P to (-1.52, 7.27) is sqrt[(-3 - (-1.52))^2 + (9 - 7.27)^2] ≈ 2.97.

Therefore, the minimum distance from P to the circle is approximately 2.97, and the maximum distance is approximately 3.89.

The number of miles travelled is 6.2 miles.

What is Kilometer ?

A kilometer (km) is a unit of length or distance measurement in the metric system. It is equivalent to 1,000 meters or approximately 0.62 miles. The prefix "kilo" means "thousand", so one kilometer is equal to 1,000 meters.

Completing the statements:

Kilometers are a larger unit of distance measurement compared to miles. That means the number of kilometers in a distance will always be greater than the number of miles in that distance.

We can convert miles to kilometers by multiplying the number of miles by 1.61. For example, if we have a distance of 5 miles, we can convert it to kilometers as:

5 × 1.61  = 8.05 kilometers

We can also convert kilometers to miles by multiplying the number of kilometers by 0.62. For example, if we have a distance of 10 kilometers, we can convert it to miles as:

10  × 0.62  = 6.2 miles

Therefore, The number of miles travelled is 6.2 miles.

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The interest rate is 4.5%, if the interest is compounded daily on an investment of $350.

To find the interest rate, compound interest formula

A= P(1+r/n)ⁿᵃ

where

A = The amount to be received

P = The Principal

r = The rate of interest

n = number of years (Here interest is compounded on daily basis. So, n =365)

a = Time period in years

Substitute the values in the formula,

790= 350(1+r/365)⁽³⁶⁵⁾⁽¹⁸⁾

790= 350(1+r/365)⁶⁵⁷⁰

790/350= (1+r/365)⁶⁵⁷⁰

Using logarithms property on both sides

ln(790/350)= 6570×ln(1+r/365)

By the property of logarithms, for small values of x ln(1+x) =x

(ln(790/350))/6570= r/365

Therefore

The rate of interest r = 4.5%

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None of the options presented represent the same function as the given equation 8x − 2y = 25.

The equation 8x − 2y = 25 represents a linear function in terms of variables x and y. To determine which equation represents the same function, we need to look for an equation that has a similar form.

A. "The number of minutes m to cook c cups of rice" does not have the same form as the given equation, so it does not represent the same function.

B. "The volume V of a cube with side length s" also does not have the same form as the given equation, so it does not represent the same function.

C. "The distance walked after m minutes at r feet per minute" does not match the given equation, so it does not represent the same function.

D. "The cost C for t tickets to a museum does not have the same form as the given equation, so it does not represent the same function.

Therefore, none of the given options represent the same function as the  equation 8x − 2y = 25.

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To write the given expression in factored form using the greatest common factor and distributive property, we need to find the largest common factor of 9 and 6, which is 3. Then we can factor out 3 from both terms, giving us 3(3d+2e). Therefore, the equivalent expression in factored form is 3(3d+2e).

This expression is simplified and shows that 3 is a common factor of both terms. In 100 words, this process involves identifying the greatest common factor between the terms and then using the distributive property to factor it out. This simplifies the expression and allows for easier calculations in further operations.

It is important to always look for common factors and simplify expressions whenever possible.

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Answer:

-4 = 5

0 = -1

2 = -4

4 = -7

(-4, 5)

(0, -1)

(2, -4)

(4, -7)

Step-by-step explanation:

First, let's identify what each term represents.

y-intercept: -1

slope: 3/2

Then, fill out the table.

x = -4

-1 - (3/2 · -4)

-1 - (-12/2)

-1 - (-6)

-1 + 6

y = 5

x = 0

-1 - (3/2 · 0)

-1 - 0

y = -1

x = 2

-1 - (3/2 · 2)

-1 - (6/2)

-1 - (3)

y = -4

x = 4

-1 - (3/2 · 4)

-1 - (12/2)

-1 - (6)

y = -7

Then, plot the points in the function on the graph.

(-4, 5)

(0, -1)

(2, -4)

(4, -7)

Finally, integrate with respect to u:

[4u](5 to 6) = 4(6) - 4(5) = 4

So, the double integral ∫∫R dydA is equal to 4.

To compute the double integral ∫∫R dydA, where D = Ф(R) and Ф(u, v) = (u, u + v), we first need to transform the integral using the given mapping.

The region R is defined as the set of all points (u, v) such that u ∈ [5, 6] and v ∈ [0, 4]. According to the transformation Ф, we have x = u and y = u + v.

Now we need to find the Jacobian determinant of the transformation:

J(Ф) = det([∂x/∂u, ∂x/∂v; ∂y/∂u, ∂y/∂v]) = det([1, 0; 1, 1]) = (1)(1) - (0)(1) = 1

Since the Jacobian determinant is nonzero, we can change the variables in the double integral using the transformation Ф:

∫∫R dydA = ∫∫D (1) dydx = ∫(5 to 6) ∫(u to u + 4) dydu

Now, compute the integral:

∫(5 to 6) ∫(u to u + 4) dydu = ∫(5 to 6) [y](u to u + 4) du
= ∫(5 to 6) [(u + 4) - u] du = ∫(5 to 6) 4 du

Finally, integrate with respect to u:

[4u](5 to 6) = 4(6) - 4(5) = 4

So, the double integral ∫∫R dydA is equal to 4.

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If ABC is similar to XYZ and YZX, then the corresponding angles of those triangles are same, and the corresponding sides are proportional. because of this triangle ABC is a special type of triangle known as a "similar triangle."

In a similar triangle, the angles of the triangle are equal, however the sides can be exclusive lengths. but, the ratios of the corresponding aspects are usually the same. This property is beneficial in lots of regions of mathematics and physics, which includes trigonometry and the study of geometric shapes.

inside the case of triangle ABC, the fact that it's far much like each XYZ and YZX tells us that its angles are same to those of these triangles, and its sides are proportional to the ones of these triangles. This property may be used to solve many issues regarding triangles and other geometric shape.

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Hence, the required equation of the quintic with double roots –4 and 2 is [tex]f(x)=\frac{1}{256} (x+4)^2(x-2)^2(x)[/tex].

Given, an equation of a quintic with double roots –4 and 2, that goes through the origin as well as (4, 4).

Let r be the remaining root of the equation.

Let the required equation in factored form is

[tex]f(x)=a(x+4)^2(x-2)^2(x-r)[/tex]

Given, the quintic goes through the origin.

Then, we know that f(0) = 0.

[tex]f(0)=a(0+4)^2(0-2)^2(0-r)[/tex]

0 = a(16)(4)(-r)

0 = -64ar

64ar = 0

either a = 0 or r = 0.

if a = 0

then the equation reduces to f(x) = 0, which is not a quintic.

a ≠ 0

This means that r = 0

So equation becomes [tex]f(x)=a(x+4)^2(x-2)^2(x)[/tex]   ...(1)

Given, the quintic goes through the point (4, 4)

So, f(4) = 4

[tex]f(4)=a(4+4)^2(4-2)^2(4)[/tex]

4 = 1064 a

a = 4/1064

a = 1/256

Putting in equation (1)

[tex]f(x)=\frac{1}{256} (x+4)^2(x-2)^2(x)[/tex]

Hence, the required equation of the quintic with double roots –4 and 2 is [tex]f(x)=\frac{1}{256} (x+4)^2(x-2)^2(x)[/tex].

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If the probability that this teacher is female is 3/5 , there are a total of 90 teachers at the school.

Let's denote the total number of teachers at the school as T. We know that the probability of choosing a female teacher is 3/5. Therefore, the probability of choosing a male teacher is 1 - 3/5 = 2/5.

We are also given that there are 36 male teachers at the school. We can use this information to set up an equation:

36/T = 2/5

To solve for T, we can cross-multiply:

36 x 5 = 2 x T

180 = 2T

T = 90

Therefore, there are a total of 90 teachers at the school.

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Complete question is:

One of the teachers at a school is chosen at random. The probability that this teacher is female is 3/5 There are 36 male teachers at the school. Work out the total number of teachers at the school.

It appears to be setting up a proportion between the central angle of the sector and the ratio of arc length to the circumference of the circle, rather than the ratio of the central angle to the full angle of the circle. Then the area should equal  to 126.9 [tex]ft^2.[/tex]

To find the area of sector XZY, we need to know the measure of angle XYZ. However, the given equation n/360 = 115/225 is incorrect, as it appears to be setting up a proportion between the central angle of the sector and the ratio of arc length to the circumference of the circle, rather than the ratio of the central angle to the full angle of the circle.

To find the correct measure of angle XYZ, we need to use the formula:

Area of sector XZY = (n/360) x π[tex]r^2[/tex]

where r is the radius of circle Z.

We know that the area of circle Z is 255 square feet, so we can find the radius as follows:

Area of circle Z = π[tex]r^2[/tex]

255 = π[tex]r^2[/tex]

[tex]r^2[/tex] = 81

r = 9

Now we can solve for n using the given ratio of 115/225:

n/360 = 115/225

n = (115/225) x 360

n = 184.32

Rounding to the nearest tenth, we get:

n ≈ 184.3

Finally, we can find the area of sector XZY as:

Area of sector XZY = (n/360) x π[tex]r^2[/tex]

Area of sector XZY = (184.3/360) x π[tex](9)^2[/tex]

Area of sector XZY ≈ 126.9 [tex]ft^2[/tex]

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The volume of small cone-shaped cups is 11.8 in³.

To find the volume of the small shake in a cone-shaped cup that is 7 inches tall and has a diameter of 3 inches, we can use the formula for the volume of a cone:

V = 1/3 πr²h

where V = volume

r = radius

h = height of the cone

Given, diameter of come is 3 inches

We know r = d/2

r = 3/2

= 1.5

Substituting the value in the formula

V = 1/3 × 3.14 × 7 × (1.5)²

= 11.78

Rounding to nearest tenth

V = 11.8

Hence, the volume of small cone-shaped cups is 11.8 in³.

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The equation y = |x - 5| + |x + 5| can be rewritten as:

y = { -2x - 10, for x < -5,

 { 10, for -5 ≤ x ≤ 5,

 { 2x + 10, for x > 5.

When -5 < x < 5, both |x - 5| and |x + 5| are non-negative. So we can rewrite y = |x - 5| + |x + 5| as follows:

If x < -5, then x - 5 < -5 and x + 5 < 0, so we have:

y = -(x - 5) - (x + 5) = -2x - 10

If -5 ≤ x ≤ 5, then x - 5 < 0 and x + 5 ≥ 0, so we have:

y = -(x - 5) + (x + 5) = 10

If x > 5, then x - 5 ≥ 0 and x + 5 > 5, so we have:

y = (x - 5) + (x + 5) = 2x + 10

Therefore, the equation y = |x - 5| + |x + 5| can be rewritten as:

y = { -2x - 10, for x < -5,

 { 10, for -5 ≤ x ≤ 5,

 { 2x + 10, for x > 5.

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Given question is incomplete, the complete question is below

Rewrite each equation without absolute value for the given conditions. y = |x-5| + |x+5| if -5 < x < 5

The p-value is 0.00012 which is less than the significance level (α = 0.1), we reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis that the true population proportion is different from 0.77.

The hypothesis being tested is:

H0: p=0.77 (null hypothesis)

H1: p≠0.77 (alternative hypothesis)

where p is the true population proportion.

The test statistic for this hypothesis test is the z-score, which can be calculated using the formula:

z = (x - np) / sqrt(np(1-p))

where x is the number of successes, n is the sample size, and p is the hypothesized proportion under the null hypothesis.

In this case, n = 500, x = 370, and p = 0.77. Plugging these values into the formula, we get:

z = (370 - 500 * 0.77) / sqrt(500 * 0.77 * 0.23)

z ≈ -3.81

The p-value for this test is the probability of obtaining a z-score more extreme than -3.81, assuming the null hypothesis is true. Since this is a two-tailed test, we need to calculate the area in both tails of the standard normal distribution. Using a standard normal distribution table or a calculator, we find that the area in each tail is approximately 0.00006.

Therefore, the p-value is:

p-value ≈ 2 * 0.00006 = 0.00012

In terms of practical interpretation, we can say that there is evidence to suggest that the proportion of successes is significantly different from 0.77 in the population from which the sample was drawn.

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The probability of P(2 | 2 of hearts) is 1/51, P(heart | 2 of hearts) is 12/51, P(club | 2 of hearts) is 13/51 and P(9 | 2 of hearts) is 4/51.

Since Sara did not replace the first card, there are now only 51 cards left in the deck, and only one of them is the 2 of hearts. Therefore, the probability that the second card is another 2 is

P(2 | 2 of hearts) = 1/51

After drawing the 2 of hearts, there are now 12 hearts left in the deck out of 51 cards. So the probability that the second card is another heart is

P(heart | 2 of hearts) = 12/51

Similarly, there are 13 clubs left in the deck out of 51 cards. So the probability that the second card is a club is

P(club | 2 of hearts) = 13/51

There are four 9s left in the deck out of 51 cards. So the probability that the second card is a 9 is

P(9 | 2 of hearts) = 4/51

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The best prediction for the bacteria population after 8 days is approximately 14,301.67 bacteria.

To find the predicted population of bacteria after 8 days, we need to apply the given growth rate of 220% per day to the initial population of 50 bacteria for each day, starting from day 1 and continuing to day 8.

For each day, the population of bacteria is expected to be 220% or 2.2 times the population of the previous day. So, we can use the formula:

P = P0 [tex]x (1 + r)^n[/tex]

where P is the predicted population after n days, P0 is the initial population, r is the growth rate per day (as a decimal), and n is the number of days.

Substituting the given values, we get:

P = 50[tex]x (1 + 2.2)^8[/tex]

P ≈ 14,301.67

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The shape of the cross section of a right rectangular pyramid sliced vertically (down) by a plane not passing through the vertex of the pyramid m is a trapezoid. (A)

This is because when a pyramid is sliced vertically, the resulting cross section is always a two-dimensional representation of the pyramid's base.

Since the base of a right rectangular pyramid is a rectangle, slicing it vertically will result in a trapezoid-shaped cross section. The top and bottom sides of the trapezoid will be parallel, and the other two sides will be slanted.

In a right rectangular pyramid, the vertex m is located directly above the center of the rectangle base. When a plane is passed through this vertex, it will result in a triangular cross section. However, when a plane is passed through a different point, as described in the question, it will result in a trapezoidal cross section.(A)

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The total surface area of the prism is 282

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

The net of a triangle prism.

The total surface area of the prism is the sum of the individual shapes

So, we have

Surface area = 2 * 1/2 * 6 * 5 + 3 * 6 * 14

Evaluate

Surface area = 282

Hence. the total surface area of the prism is 282

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Answer:

The total surface area is 282

The value of the expression  a+b = 45+8 = 53

To write (3x)² (5x⁶) in the form of ax^b, we need to simplify the expressions and multiply the coefficients and the variables separately:

(3x)² (5x⁶) = 9x² × 5x⁶ = 45x^(2+6) = 45x^8

So, the expression (3x)² (5x⁶) can be written as 45x^8 in the form of ax^b, where a=45 and b=8.

Therefore, a+b = 45+8 = 53

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If at the previous appointment the baby weighed 10 pounds and 8 ounces, then the baby has gained 12 ounces since the last appointment.

To calculate this, we need to subtract the weight at the previous appointment from the weight at the current appointment:

11 pounds and 4 ounces - 10 pounds and 8 ounces = 12 ounces

So the baby has gained 12 ounces since the last appointment. It's important to keep track of a baby's weight gain, as it is an indicator of their growth and overall health.

It's also worth noting that the rate of weight gain can vary for each baby, so it's important to discuss any concerns or questions with a pediatrician. Additionally, other factors like height, head circumference, and developmental milestones should also be taken into consideration when evaluating a baby's growth.

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The perimeter, P, of the hexagon in terms of x is 18x + 24.

To find the perimeter, P, of the hexagon in terms of x, we'll consider the given side lengths.

The hexagon has 4 sides of length 3x + 5. The other 2 sides are each 3 units shorter than the other 4 sides, so their length is (3x + 5) - 3 = 3x + 2.

Now, we can calculate the perimeter by adding the lengths of all 6 sides:

P = (4 * (3x + 5)) + (2 * (3x + 2))

First, distribute the numbers to the expressions inside the parentheses:

P = (12x + 20) + (6x + 4)

Next, combine like terms:

P = 18x + 24

So, the perimeter, P, of the hexagon in terms of x is 18x + 24.

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Hence, the correct option is C.

An equation is a mathematical statement that shows the equality between two expressions.

1. Twice the sum of four and a number can be written as 2(4 + x), where x is the number.

2. The sum of a number and 32 can be written as x + 32, where x is the number.

3. Five is half of a number and 32 can be written as 5 = 0.5x + 32, where x is the number.

To see why, we can use the fact that "half of a number" can be written as 0.5x, so the sentence becomes 5 = 0.5x + 32 and hence become equation.

4.The quotient of 15 and a number can be written as 15/x, where x is the number.

Therefore, 5 = 0.5x + 32, which can be simplified to 0.5x = -27, and then to x = -54.

Hence, the correct option is C.

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The value of the quantity after 51 years, rounded to the nearest hundredth, is 499.92.

Since a decade is a period of 10 years, a decay rate of 5% per decade can be converted to a continuous decay rate as follows:

Continuous decay rate = (1 + decay rate per decade[tex])^{(1/10)[/tex] - 1

In this case, the decay rate per decade is 5%, which can be expressed as 0.05.

Continuous decay rate = (1 + 0.05[tex])^{(1/10)[/tex] - 1

Continuous decay rate ≈ 0.0048767

Now we can use the formula for continuous decay:

A = A0[tex]e^{rt[/tex]

In this case, the initial value A0 is 390, the continuous decay rate r is 0.0048767, and the time elapsed t is 51 years.

Substituting these values into the formula, we have:

A = 390 [tex]e^{(0.0048767)( 51)[/tex]

A ≈ 499.9202826

Therefore, the value of the quantity after 51 years, rounded to the nearest hundredth, is 499.92.

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A geometric sequence is a type of sequence where each term is found by multiplying the previous term by a constant factor. This means that each term is a multiple of the one before it. In contrast, an arithmetic sequence is a type of sequence where each term is found by adding a constant value to the previous term.

This means that each term is a sum of the one before it and a fixed value.

To answer your question, whether a geometric sequence will always grow faster than an arithmetic one depends on the values of the constant factor and fixed value in each sequence. In general, if the constant factor in a geometric sequence is greater than 1, the terms will grow at an increasingly faster rate than in an arithmetic sequence.

However, if the constant factor is between 0 and 1, the terms will grow at a decreasing rate, meaning that the sequence will actually grow more slowly than an arithmetic one.

It's important to note that the rate of growth is not the only factor to consider when comparing geometric and arithmetic sequences. The actual values of the terms in each sequence can also differ significantly, depending on the starting term and the values of the common ratio and common difference.

In some cases, an arithmetic sequence may actually have higher values than a geometric one, even if it grows more slowly.

In summary, whether a geometric sequence will always grow faster than an arithmetic one depends on the specific values of each sequence. However, in general, if the constant factor in a geometric sequence is greater than 1, it will grow faster than an arithmetic sequence.

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