3. A savings account is started with an initial deposit of $500. The account earns 4.25% interest compounded annually.

(a) Write an equation to represent the amount of money in the account as a function of time in years.

(b) Find the amount of time it takes for the account balance to reach 1 million. Show your work.
Note: 1 million is a 1 with 6 zeros.

Note2: you must use log functions to solve as Unit6 is about using log functions.
Answer:

The maximum productivity is 2646 units per month and it occurs at 21 years of service.

To find the maximum productivity and the year in which it is achieved, we need to find the vertex of the parabolic function represented by the productivity function.

The vertex of a parabola in the form y = a[tex]x^{2}[/tex] + bx + c is located at the point (-b/2a, c - [tex]b^{2}[/tex]/4a).

In this case, we have:

M(t) = -3[tex]t^{2}[/tex] + 126t + 150

So, a = -3, b = 126, and c = 150.

The year of service, t, is restricted to the interval 0 ≤ t ≤ 40, which means that the maximum productivity will occur at either t = 0 or t = 40, or at the vertex of the parabola.

The x-coordinate of the vertex is given by -b/2a, which in this case is:

t = -b/2a = -126/(2(-3)) = 21

So, the maximum productivity occurs at t = 21 years of service.

To find the maximum productivity, we substitute t = 21 into the productivity function:

M(21) = -3[tex](21)^{2}[/tex] + 126(21) + 150 = 2646

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The equation of the line having slope 0 and passing through (-3,-2) is f(x) = -2.

Any horizontal line has the same y-value for every point on the line. We are given that the line passes through the point (-3,-2). This means that f(-3) = -2, since the y-value of the point (-3,-2) corresponds to the value of the function at x = -3.

This is because no matter what x-value we plug into the function, the output (y-value) will always be -2. Therefore, the equation of the line in function notation is f(x) = -2.

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The surface area of the two hemispheres combined is 100π square meters, and the volume of the two hemispheres combined is (500/3)π cubic meters.

The circumference of a great circle on a sphere is given by the formula C = 2πr, where r is the radius of the sphere.

In this case, we are given that the circumference of the great circle is 10π meters, so we can set up an equation:

10π = 2πr

Dividing both sides by 2π, we get:

r = 5

So the radius of the sphere is 5 meters.

When the sphere is cut in half, we get two equal hemispheres. The surface area of a closed hemisphere is given by the formula A = 2πr^2, where r is the radius of the hemisphere.

Since we are dealing with a radius of 5 meters, the surface area of one hemisphere is:

A = 2π(5^2) = 50π

Therefore, the surface area of the two hemispheres combined is:

2A = 2(50π) = 100π

The volume of a closed hemisphere is given by the formula V = (2/3)πr^3. Again, since we are dealing with a radius of 5 meters, the volume of one hemisphere is:

V = (2/3)π(5^3) = (2/3)π125 = (250/3)π

Therefore, the volume of the two hemispheres combined is:

2V = 2(250/3)π = (500/3)π

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Answer:

  B.  quadratic

Step-by-step explanation:

You want to know the kind of function that would best fit the points shown on the graph.

The points on the graph demonstrate a slope that starts out positive and decreases through 0 to a slope with a negative value.

A linear function has a constant slope.

An exponential function has a continuously increasing or continuously decreasing slope.

A quadratic function has a slope that changes sign, matching the slope of the curve joining the given points.

The equation that best fits the given data will be quadratic.

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The figure of the shaded region is shown in graph.

Now, Based on the given information, we can create the following system of inequalities to represent Josh's weekly exercise program:

for he exercises for at least 10 hours;

⇒ x + y ≥ 10

And, When he spends at most 8 hours doing cardiovascular work is,

x ≤ 8

When, he spends at most 6 hours on weight training,

y ≤ 6

Hence, To graph these inequalities, we can start by graphing the three lines that represent each of the above equations.

The first inequality, x + y ≥ 10, represents a line with a y-intercept of 10 and a slope of -1.

The second inequality, x ≤ 8, represents a vertical line at x = 8.

The third inequality, y ≤ 6, represents a horizontal line at y = 6.

Thus, The figure of the shaded region is shown in graph.

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Answer:

Hi

You can identify which is the shortest and longest by converting the fractions to decimal or by finding the L.C.M

The longest time is 1

and the shortest is 1/4

The difference would be 1-1/4= 3/4

7 groups of 2/7 is equivalent to the whole, which is represented by the number 2.

When we have 7 groups of 2/7, we are essentially multiplying the fraction 2/7 by the whole number 7. To perform this operation, we can use the following formula:

a/b × c/d = ac/bd

where a, b, c, and d are any numbers, and a/b and c/d are two fractions.

In our case, a is 2, b is 7, c is 7, and d is 1. This is because we want to multiply 2/7 by 7 to get 7 groups of 2/7. Therefore, we can substitute these values into the formula:

2/7 × 7/1 = (2 × 7) / (7 × 1) = 14/7

The result is a fraction 14/7, which can be simplified to 2. This means that 7 groups of 2/7 is equal to 2. To simplify this further, we can say that if we divide a whole into 7 equal parts, each part is 2/7. If we have 7 of these parts, we have the whole.

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Answer:

Question 5 : 10% decay

Question 6 : Amar is right

Step-by-step explanation:

The length of the tree line, t, to the nearest foot, is 20 ft.

We have,

From the figure,

t can be considered as the hypotenuse.

And,

The other two sides are::

15 - (7 + 5) = 15 - 12 = 3

30 - 10 = 20

Now,

Applying the Pythagorean theorem,

t² = 3² + 20²

t² = 9 + 400

t² = 409

t = 20

Thus,

The length of the tree line, t, to the nearest foot, is 20 ft.

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Answer:

Fairview and Union are 9 cm apart on the map, and the scale is 1 cm: 10 km. So, to figure out the actual distance between the two cities, we need to use some math.

If 1 cm on the map represents 10 km in real life, then 9 cm on the map represents 90 km in real life. Therefore, Fairview and Union are actually 90 km apart.

Hope that helps!

Step-by-step explanation:

Answer:

90 km

Step-by-step explanation:

The scale of the map is

1 cm : 10 km

The scale of a map is a ratio. We can use a proportion to solve this problem.

1 cm is to 10 km as 9 cm is to x km

1/10 = 9/x

1 × x = 10 × 9

x = 90

Answer: 90 km

The value of the variable x = 21/32

Algebraic expressions are simply described as expressions that are composed of variables, coefficients, terms, constants and factors.

These expressions are also made up of certain arithmetic or mathematical operations.

These operations includes;

From the information given, we have that;

3/8=4/7x

cross multiply the values, we have;

7x(3) = 8(4)

multiply the values and expand the brackets, we get the values;

21x = 32

Divide both sides by the coefficient of the variable x, we get;

x = 32/21

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Answer:

A

Step-by-step explanation:

0.15 × 12 = 1.8

ps : i'm not sure if it's true or not, but i think this is the answer. i'm sorry if i get you wrong

The probability that that arrow will land on the part labelled C after the first spin would be = 1/5.

To calculate the probability of the chosen event, the formula for probability should be used which is given as follows:

Probability = possible outcome/ sample space

Where possible outcome = 2

sample space = A,B,B,B,C,C,D,D,D,E = 10

Therefore the probability that the arrow will land on the part labelled C after the first spin = 2/10 = 1/5.

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The point estimate of the population mean is 10.5.

The point estimate of the population standard deviation is 3.240, rounded to 3 decimals.

The margin of error for the estimation of the population mean with 95% confidence is 3.1

The 95% confidence interval for the population mean is [7.4, 13.6]

a. The point estimate of the population mean is the sample mean.

We can calculate it by adding up all the sample values and dividing by the sample size:

(10 + 8 + 12 + 15 + 13 + 11 + 6 + 5) / 8 = 10.5

Therefore, the point estimate of the population mean is 10.5.

b. The point estimate of the population standard deviation is the sample standard deviation. We can calculate it using the following formula:

s = √Σ(xi - x)² / (n - 1) ]

Plugging in the sample values, we get:

x = 10.5

s = √((10-10.5)² + (8-10.5)² + (12-10.5)² + (15-10.5)²+ (13-10.5)² + (11-10.5)²+ (6-10.5)² + (5-10.5)²) / (8 - 1) ]

s = 3.240

c.

margin of error = t-value × (sample standard deviation /  √(sample size))

Plugging in the values we found, we get:

margin of error = 2.365 × (3.240 /  √(8))

margin of error=3.067

d. Finally, we can use the following formula to find the 95% confidence interval for the population mean:

confidence interval = sample mean ± margin of error

confidence interval = 10.5 ± 3.067

confidence interval = [7.4, 13.6]

Therefore, the 95% confidence interval for the population mean is [7.4, 13.6]

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The number of hours that Konner is left with till they can reach their destination is 1 ¹ / ₃ hours

We can start by using a proportion to set up an equation to solve for the time needed to arrive:

Distance traveled / total distance = time traveled / total time

Using a ratio, we infer that 5 / 6 of an hour is equivalent to 1 / 3 of the way.

This means that if there are 2 / 3 of the way left, the hours left till Konner reaches is:

= 5 / 6 x 2

= 10 / 6

= 5 / 3

= 1 ¹ / ₃ hours

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Question is:

How much longer till he arrives ?

The required graph has been attached below that represents the inverse equation f(x) = 4x-1.

To graph the inverse equation f(x) = 4x - 1, we can follow these steps:

Replace f(x) with y:

y = 4x - 1

Switch the x and y variables: x = 4y - 1

Solve for y:

y = (x + 1) / 4

The graph of the original function f(x) = 4x - 1 is a line with a slope of 4 and a y-intercept of -1.

The graph of the inverse function is the reflection of this line over the line y = x, which means that the x and y coordinates of each point are swapped.

To plot points for the inverse function, we can choose some values of x and find the corresponding values of y using the equation y = (x + 1) / 4. For example:

x y

-3 -1/4

-2 0

-1 1/4

0 1/4

1 1/2

2 3/4

3 1

Therefore, the graph of the inverse equation f(x) = 4x - 1 is a line that passes through the points (-3, -1/4), (-2, 0), (-1, 1/4), (0, 1/4), (1, 1/2), (2, 3/4), and (3, 1).

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Answer:

(4,0) & (9,5)

Step-by-step explanation:

Where the equations cross are the solutions!

Hope this Helps! :)
Happy Studies

The complement of a set C, denoted by C', contains all the elements of the universal set that are not in C. In this case, the universal set is A, which contains all the flavors and toppings available.

Given data ,

a) {vanilla, strawberries, sprinkles}

This set contains vanilla, strawberries, and sprinkles but does not contain any elements from C. Therefore, C' = A - {vanilla, strawberries, sprinkles} = {chocolate, hot fudge, peanuts, whipped cream}.

b) {chocolate, hot fudge, peanuts, whipped cream}

This set contains all the elements of C but does not contain any elements from A that are not in C. Therefore, C' = A - C = {vanilla, strawberries, sprinkles}.

c) {vanilla, hot fudge, sprinkles, whipped cream}

This set contains hot fudge, sprinkles, and whipped cream but does not contain any elements from C. Therefore, C' = A - {hot fudge, sprinkles, whipped cream} = {vanilla, chocolate, strawberries, peanuts}.

d) {chocolate, strawberries, sprinkles}

This set contains strawberries and sprinkles but does not contain any elements from C. Therefore, C' = A - {strawberries, sprinkles} = {vanilla, chocolate, hot fudge, peanuts, whipped cream}.

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The expression is equivalent to 10w - 25 = 0

It is important to note that algebraic expressions are described as expressions that are made up of terms, variables, coefficients, factors and constants.

These expressions are also made up of arithmetic operations. These operations are;

From the information given, we have that;

D =  2w - w - 7

C = 3w - 6

To determine the value of the expression  D + 3C

substitute the expression

2w - w - 7 + 3(3w -6)

expand the bracket

2w - w - 7 + 9w - 18

collect the like terms and add

10w - 25 = 0

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If a line has a slope of 3 and contains the point (-1, 4), its equation in point-slope form include the following: C. y - 4 = 3(x - 1).

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁)

Where:

At data point (1, 4) and a slope of 3, a linear equation for this line can be calculated or determined by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 4 = 3(x - 1)

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Answer: x = 15

Step-by-step explanation:

Answer:

the answer for 2/3(x-6) = 6. what's x? is 15

Answer:

Option 2 and Option 3 are equivalent to 5-3 × [10x (6-2)]

Step-by-step explanation:

Option 2:

5-3 × [10x (6-2)] = 5-3 × [10x4] = 5-3 × 40 = 5-120 = -115

Option 3:

5-3 × [10x (6-2)] = 5-3 × 40 = 5-120 = -115

The standard deviation of the given data is approximately 3.08.

To find the standard deviation of the given data (5, 6, 8, 13), we first need to find the mean (X) of the data.

Mean (X) = (5 + 6 + 8 + 13) / 4 = 32 / 4 = 8

Next, we calculate the deviations from the mean for each data point:

Deviation from the mean = Data point - Mean

For 5: 5 - 8 = -3

For 6: 6 - 8 = -2

For 8: 8 - 8 = 0

For 13: 13 - 8 = 5

Then, we square each deviation:

(-3)^2 = 9

(-2)^2 = 4

0^2 = 0

5^2 = 25

Next, we calculate the sum of the squared deviations:

Sum of squared deviations = 9 + 4 + 0 + 25 = 38

To find the variance, we divide the sum of squared deviations by the number of data points:

Variance = Sum of squared deviations / Number of data points = 38 / 4 = 9.5

Finally, we take the square root of the variance to find the standard deviation:

Standard deviation = √Variance = √9.5 ≈ 3.08

Therefore, the standard deviation of the given data is approximately 3.08.

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Answer: -4, -2 and 5, 7

Step-by-step explanation:

Answer:

The answer is 43°

Step-by-step explanation:

137+137+x+43=360°

x+317=360

x=360-317

x=43°

The number of square meters that the space that Araceli rents is, is 30. 66 square meters.

The space that Araceli rents has a rectangular space so to find the area of the rectangular space that Araceli rents, we need to multiply the length by the width.

The given width is 4. 2 meters and the length is 7. 3 meters.

The area of the space in square meters is therefore :

Area = Length × Width

Area = 7. 3 meters × 4. 2 meters

Area = 30. 66 square meters

In conclusion, the space that Araceli rents has an area of 30. 66 square meters.

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The probability that a point falls in the red-shaded square is given as follows:

p = 9/16.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The probability that a point falls in the red shared square is then given by the division of the area of the red shaded square by the division of the area of the larger square.

The area of the red-shaded square is given as follows:

3² = 9.

The area of the larger square is given as follows:

4² = 16.

Hence the probability is given as follows:

p = 9/16.

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The solution of the fraction is n = 2 (option c).

Now, let's consider the equation n/32 = 1/16, where n is a variable that we want to solve for. This equation tells us that the ratio of n to 32 is equal to the ratio of 1 to 16. In other words, if we divide n by 32, we get the same result as if we divide 1 by 16.

To solve for n, we can use algebraic techniques to isolate the variable on one side of the equation. One way to do this is to multiply both sides of the equation by 32, which gives us:

n = (1/16) * 32

We can simplify the right-hand side of this expression by multiplying 1/16 by 32, which gives us:

n = 2

Therefore, the answer to this problem is n = 2. (option c).

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Complete Question:

n/32 = 1/16. Fine the value of n.

a. n = one-half

b. n = 4

c. n = 2

d. n = 512