Let f(t) be the population (in thousands of people) of Math City, as a function the time t (in years), where t = 0 corresponds to the year 2020. In 2020, the population was 34000, and each year the population increases by 3000 people.
(a) Model the population by finding a formula for f(t), using the appropriate choice of a linear, polynomial, power, trigonometric, or exponential function, or a piecewise combination thereof.
(b) Evaluate the expression f(6), and write a sentence explaining what this means.
(c) Solve the equation f(t) = 64, and write a sentence explaining what this means

Answer: We can start by using the concept of rates.

At 12:00 noon, oil is poured into the top of the tank at a rate of 2.1 gallons per minute and the valve at the bottom empties at a rate of 1.9 gallons per minute.

As the tank is empty at the beginning, the rate at which the oil is poured into the tank is equal to the rate at which the oil is leaving the tank.

2.1 gallons per minute - 1.9 gallons per minute = 0.2 gallons per minute

As the tank does not fill until 12:00 the next day, the difference between the rate of oil that enters the tank and the rate of oil that leaves the tank is equal to zero over the time frame.

0.2 gallons per minute * 1440 minutes = 288 gallons

So, the tank holds 288 gallons.

Step-by-step explanation:

The position u of the mass at any time t is u(t) = 0.008sin(25t) m, and the mass first return to its equilibrium position when t = π/25 s.

m = 0.25 kg

L = 0.01568 m

u(0) = 0, u′(0) = 0.2 m/s, where u(t) is position of the mass at any time t.

k = mg/L

= (0.25×9.8)/0.01568

= 156.25 N/m

Now we can write the equation of the system: mu′′ + ku = 0

0.25u′′ + 156.25u = 0

u′′ + 625u = 0

The characteristic equation: r² + 625 = 0,  r = ±25i .

The general solution is u(t) = C₁sin⁡(25t) + C₂cos⁡(25t)

u(0) = 0 = C₁ × 0 + C₂ × 1 = C₂, C2​=0

u′(0) = 0.2 = 25C₁, C₁ = 0.008

So, u(t) = 0.008sin(25t) m.

Equilibrium position: u(t)=0,  0.008sin⁡(25t)=0,  sin⁡(25t)=0,  25t=πn, n∈Z

The mass first returns to its equilibrium position when n = 1 or t = π/25.

--The given question is incorrect; the correct question is

"A mass of 250 g stretches a spring 1.568 cm. If the mass is set in motion from its equilibrium position with a downward velocity of 20 cms, and if there is no damping, determine the position u of the mass at any time t.

Enclose arguments of functions in parentheses. For example, sin(2x).

a. Assume g = 9.8 m/s^2. Enter ans exact answer.

u(t) = ______ m

b. When does the mass first return to its equilibrium position?

Enter an exact answer.

t = ______s"--

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Answer:

Option A) [tex]132ft[/tex]

Step-by-step explanation:

It is given that,

Radius of a circle = 21ft

We know that π = [tex]\frac{22}{7}[/tex]

So the circumference of a circle = 2πr

Substituting the values,

= [tex]2X\frac{22}{7} X 21[/tex]

So we get,

[tex]= 2X22X3[/tex]

[tex]= 132ft[/tex]

The required circumference of the circle is 132 feet.  Option A is correct.

To find the circumference of a circle, we use the formula C = πd, where C is the circumference, π is the mathematical constant pi (approximately equal to 3.14), and d is the diameter of the circle.

Here,
The circumference of a circle with a radius of 21 feet can be found using the formula:

C = 2πr

where "C" is the circumference, "r" is the radius, and "π" is the mathematical constant pi, which is approximately equal to 22/7.

Substituting the values given, we have:

C = 2 x (22/7) x 21

C = 132 feet

Therefore, the circumference of the circle is 132 feet. Answer: 132 feet.

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Answer:

Step-by-step explanation:

1/3 is 2/6

2/6 + 1/6 = 3/6

3/6 = 1/2

answer 1/2 or half

Answer:

It is 1/2

Step-by-step explanation:

You make 1/3 the same denominator so it becomes 2/6.

2/6+1/6=3/6

Now when you simplify this you get 1/2.

The measure of the length HL will be 18.5 units for the given figure.

One of the earliest areas of mathematics is geometry, along with arithmetic. It is concerned with spatial characteristics like the separation, shape, size, and relative placement of objects.

Given that in the quadrilateral the two sides are Jk = 12 and GF = 25. The value of HL can be calculated by the theorem,

HL = ( 1 / 2 ) x GF x JK

HL = ( 1 / 2 ) x 25 x 12

HL = 18.5

Therefore, the length of section HL will be 18.5 units.

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Answer: When a polynomial is divided by x-1, the quotient is of the form of ax^2 + bx + c and the remainder is of the form of r. The quotient and the remainder are related by the formula:

f(x) = (x-1)(ax^2 + bx + c) + r

Given that the quotient is 1x^2+x-8 and the remainder is -2, we can substitute these into the above equation:

f(x) = (x-1)(1x^2+x-8) + (-2)

f(x) = x^3 - 9x^2 + (1-8)x +(-2)

f(x) = x^3 - 9x^2 - 7x - 2

Therefore the function f(x) is x^3 - 9x^2 - 7x - 2 in standard form.

Step-by-step explanation:

Using algebraic expressions, The largest possible frame width x is 5.

When addition, subtraction, multiplication, division, and other mathematical operations are performed on variables and constants, the result is a mathematical statement known as an algebraic expression.

Let's say James and Natalie were playing with matchsticks when they had the idea to create number patterns with them.

James made the number 4 with four matchsticks. In order to create a pattern with two 4s, Natalie added three extra matchsticks. They understood that they could keep adding three matchsticks each round to make an additional "four" by doing so. They deduced from this that, generally speaking, in order to create a pattern with n number of 4s, you need 4+ 3(n-1) sticks. We refer to 4+ 3(n-1) as an algebraic expression in this case.

Let the border's width be x, making the overall frame equal to (10 + 2x)(10 + 2x).

area of boundary hence is (10+2x)(10+2x) - 100

but (10+2x)(10+2x) - 100 = 300

(10+2x)² = 400

10+2x = 20

2x = 10

x = 5

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The estimate of the total number of computers that needed repairing last year in Point City is given as follows:

Between 132,000 and 328,000 computers.

The estimate is obtained applying the proportions in the context of this problem.

The confidence interval is given as follows:

1.15 plus/minus 0.49.

Hence the bounds of the interval are given as follows:

Hence, out of 200,000 households, the lower bound of the estimate is given as follows:

0.66 x 200,000 = 132,000.

The upper bound is of:

1.64 x 200,000 = 328,000.

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Distribution (1) and (2) can be compared by looking at their medians and IQRs. If the medians and IQRs are different, the distributions can be assumed to be different.

When comparing two distributions, one of the most useful metrics is their medians and IQRs. The median is the midpoint of the data set, while the IQR is the difference between the first quartile and the third quartile. If the medians and IQRs are different, it suggests that the two distributions are different. For example, if one distribution has a lower median than the other, it suggests that the majority of data points in the lower distribution are lower than the majority of data points in the higher distribution. Similarly, if the IQR of one distribution is greater than the other, it suggests that the data points in the higher distribution are spread further apart than the data points in the lower distribution. Comparing the medians and IQRs of two distributions can help to determine if the distributions are different

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Answer: The formula to calculate the future value of an investment account is FV = PV(1+i)^n, where FV is the future value, PV is the present value, i is the interest rate (in decimal form), and n is the number of compound periods.

Given that PV = $50,000, i = 10%, and n = 4, we can plug these values into the formula:

FV = $50,000(1+0.1)^4

By using the formula, FV = $50,000(1.1)^4 = $50,000(1.4641) = $73,205.50

The future value of the retirement account after 4 years is $73,205.50

Step-by-step explanation:

The resulting augmented matrix is:

1 2 -6 | -5

0 1 -8 | 13

0 0 42 | -177

To perform the three-row operations on the augmented matrix A, we proceed as follows:

(a) R = 1 + 12:

Add 12 times row 1 with row 2,

1 2 -6 | -5

0 -14 -32 | 14

-3 -10 -2 | -4

(b) Rg - 3r + r:

Add 3 times row 2 to row 3, and subtract row 3 from row 1

1 2 -6 | -5

0 -14 -32 | 14

0 -16 -38 | -9

(c) Rs = 4r2 + 13:

Multiply row 2 by 4 and add 13 times row 2 to row 3

1 2 -6 | -5

0 1 -8 | 13

0 0 42 | -177

So the resulting augmented matrix is:

1 2 -6 | -5

0 1 -8 | 13

0 0 42 | -177

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A multiplication equation to show an equivalent fraction of 15 using fifteenths would gives 10/75

A fraction consisting of a quotient and remainder is a mixed fraction. we can convert the mixed fraction to improper fraction by first dividing the numerator by denominator and then taking the quotient as whole number and remainder as the numerator of proper fraction keeping the denominator same.

We have to write an equivalent fraction of 15 using fifteenths.

2/15

So let multiply by 5 .

2 / 15 x  5/ 5 = 10/75

Thus, 10/75 is equivalent expression.

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Answer:

0.43, 43%

Step-by-step explanation:

a. The value of k from augmented matrix [tex]\left(\begin{array}{ccc}3&- 5&k\\- 9&15&3\end{array}\right)[/tex] is - 1.

b. The value of k from augmented matrix [tex]\left(\begin{array}{ccc}k&1&- 6\\5&- 1&6\end{array}\right)[/tex] is - 5.

The complete question is in the attachment. The augmented matrix is a consistent linear system if

a. The augmented matrix [tex]\left(\begin{array}{ccc}3&- 5&k\\- 9&15&3\end{array}\right)[/tex]

Making the second row become zero R₂ = R₂ + 3R₁

[tex]\left(\begin{array}{ccc}3&- 5&k\\- 9 \:+\: (3 \times 3)&15 \:+\: (3 \times - 5)&3 \:+\: (3 \times k)\end{array}\right)[/tex]

[tex]\left(\begin{array}{ccc}3&- 5&k\\- 9 \:+\: 9&15 \:+\: (- 15)&3 \:+\: 3k\end{array}\right)[/tex]

[tex]\left(\begin{array}{ccc}3&- 5&k\\0&0&3 \:+\: 3k\end{array}\right)[/tex]

From the second row

0x + 0y = 3 + 3k

For the system to be consistent, then

3 + 3k = 0

3k = - 3

k = - 3/3

k = - 1

b. The augmented matrix [tex]\left(\begin{array}{ccc}k&1&- 6\\5&- 1&6\end{array}\right)[/tex]

Making the second row become zero R₂ = R₂ + R₁

[tex]\left(\begin{array}{ccc}k&1&- 6\\5 \:+\: k&- 1 \:+\: 1&6 \:+\: (- 6)\end{array}\right)[/tex]

[tex]\left(\begin{array}{ccc}k&1&- 6\\k \:+\: 5&0&0\end{array}\right)[/tex]

From the second row

(k + 5)x + 0y = 0

For the system to be consistent, then

k + 5 = 0

k = - 5

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The probability that the first card isn’t even number and the second card is less than five is given as follows:

1/4 = 0.25 = 25%.

A probability is obtained with the division of the number of desired outcomes by the number of total outcomes.

The probabilities for each event in this problem are given as follows:

The events are independent, hence the probability is obtained with the multiplication of the two probabilities, as follows:

1/2 x 1/2 = 1/4 = 0.25 = 25%.

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Answer:  Option (E)

4,  4√3

===========================================================

Reason:

The hypotenuse is 8. Half of that is 8/2 = 4, which is the value of "s".

This is the short leg of the right triangle. The short leg is opposite the 30 degree angle. Smallest side opposite the smallest angle.

Then use this 30-60-90 triangle template formula

[tex]\text{long leg} = (\text{short leg})*\sqrt{3}\\\\\text{long leg} = 4\sqrt{3}\\\\[/tex]

Answer:

The length is 273 and the width is 39

Step-by-step explanation:

Given:

-The perimeter is 312 feet

-The length is seven times the width

Solution:

Let's say the width is x and the length is y.

So our equation is x + y = 312

But, 7x = y

If we substituted that into our equation, it would be:

7x + x = 312,

8x = 312

Divide by 8 on both sides,

x = 39

y = 273

So the length is 273 and the width is 39

For any k ∈ (-∞, o), function f is strictly decreasing on the interval (-1, 1).

In mathematics, a function is a relationship or mapping between two sets of elements, known as the domain and the codomain, that assigns each element from the domain to a unique element in the codomain. It describes how inputs from the domain are transformed or related to corresponding outputs in the codomain.

The given function is f = [tex]\frac{kx}{x^2+1}[/tex] ...(1),

Differentiate (1) w.r.t. x,

[tex]\dfrac{d}{dx}f = \dfrac{d}{dx}\frac{kx}{x^2+1}[/tex]

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

Since f is decreasing on (-1, 1),

therefore

f' < 0 on (-1, 1),

[tex]k\frac{1-x^2}{(x^2 + 1)^2}[/tex] < 0 on (-1, 1).

As [tex]\frac{1-x^2}{(x^2 + 1)^2}[/tex] > 0 on (-1, 1),

k < 0 for f' < 0.

Therefore, k ∈ (-∞, o) for f to be strictly decreasing on (-1, 1).

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The complete question is as follows:

Let f be the function given by [tex]\frac{kx}{x^2+1}[/tex], where k is a constant. For what values of k, if any, is f strictly decreasing on the interval (-1, 1)?

The length in yards, of the carpet needed for this room is 128 yards.

length of the room = 32 ft

breadth of the room = 27 ft

so we find area now,

area = length X breadth

area = 32 X 27

area = 864ft^2

The width of the carpet in feet is = 2.25ft

so , to find the length of the carpet needed, you must apply the formula for calculate the area of a rectangle and solve for the length, as following,

area = length X width

864 = length X 2.25

length = 864/ 2.25

length = 384ft

length = 128yards

so the length, in yards, of the carpet needed for this room is 128 yards.

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Answer:

[tex]a_{n}[/tex] = 3n + 12

Step-by-step explanation:

the nth term ( explicit formula ) of an arithmetic sequence is

[tex]a_{n}[/tex] = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

here a₁ = 15 and d = a₂ - a₁ = 18 - 15 = 3 , then

[tex]a_{n}[/tex] = 15 + 3(n - 1) = 15 + 3n - 3 = 3n + 12

Answer:

0.125 pounds of skittles

Step-by-step explanation:

Lauren and Shayla have a total of ½ pound of skittles and they want to share them with 2 friends, including themselves, so the total number of people is 2+2 = 4

If they want to divide the ½ pound of skittles equally among 4 people, each person will receive ½ pound / 4 = 0.125 pounds of skittles.

The area of the circle is πr², where r is the radius. The area of the shaded region is half the area of the circle.

The area of the circle is πr², where r is the radius. The area of the shaded region is half the area of the circle.

Therefore, Area of the shaded region = πr²/2

Given that minor arc AD measures 2π, the radius of the circle is 2.

Therefore, Area of the shaded region = π(2)²/2 = 4π = 72

The area of the shaded region is half the area of the circle, which is πr²/2. The minor arc AD measures 2π, so the radius of the circle is 2, and the area of the shaded region is 4π, or 72.

The area of the circle is πr², where r is the radius. The area of the shaded region is half the area of the circle.

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Answer:

m=21/8

Step-by-step explanation:

5 /38+m=8
Move the constant to the right.
43/8+m=8

m=8-43/8

m=21/8

The area and heights  and bases of the figures are: 8.9, 54 in², 35cm²,20.25m², 168 ft², 6cm, 37.8in² and 5mm respectively

The figures include triangles and trapezoids.  The areas are the floor spaces the figures occupy

1) Area of a triangle = 1/2 *b*h

b = 14.3*2 = 28.6

h=8.9

Area = 1/2 *28.6 * 8.9

Area = 127.27 cm²

2) Area of a trapezoid = 1/2(a+b)h

Area = 1/2 (12+6)*6

Area = 1/2*18*6

= 54in²

3) Area of a trapezoid = 1/2(a+b)h

Area = 1/2 (8+6)*5

= (14*5)/2 = 35cm²

4) Area of a triangle = 1/2 *b*h

Area = 1/2 * 9 * 4.5

=20.25m²

5) Area of a trapezoid = 1/2(a+b)h

Area = 1/2 (18+24)*8

= 168ft²

6) Area of a triangle = 1/2 *b*h

38.4 = 1/2*b*12.8

Making b the subject of the relation we have

76.8 = 12.8b

b=6cm

7) Area of a triangle = 1/2 *b*h

Area = 1/2*8.4*9

=37.8in²

8) Area of a trapezoid = 1/2(a+b)h

35=1/2(6+8)*h

70=14h

Making h the subject we have

h= 5 mm

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As per the concept of algorithm, the reasonable number of steps is 2n.

The term algorithm in math is defined as the procedure for solving a mathematical problem in a finite number of steps that frequently involves repetition of an operation.

Here we have given that an algorithm has n number of steps.

Here we need to find the reasonable number of steps.

While we know the definition of algorithms, then her we need to understand the term reasonable time that is defined as  the algorithms with a polynomial efficiency or lower are said to run in a reasonable amount of time.

So, the reasonable number of steps is 2n where n refers the number of steps.

Complete Question:

If an algorithm has n number of steps, then in find out the reasonable number of steps?

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Answer:  To find the mean test score, we need to calculate the sum of the product of each score and its corresponding frequency, and divide it by the total number of students.

To estimate the mean mathematics test score, we can use the midpoint of each data class as the value for each score and multiply it by the corresponding frequency.

For the first class, 550 to 599, the midpoint is (550+599)/2 = 574.5

For the second class, 600 to 649, the midpoint is (600+649)/2 = 624.5

For the third class, 650 to 699, the midpoint is (650+699)/2 = 674.5

For the fourth class, 700 to 749, the midpoint is (700+749)/2 = 724.5

For the fifth class, 750 to 799, the midpoint is (750+799)/2 = 774.5

The mean test score is:

(5574.5 + 11624.5 + 14674.5 + 12724.5 + 7*774.5) / 49 = 669.3

Therefore, the estimate of the mean mathematics test score for the students in the study is 669.3.

Step-by-step explanation:

Answer:

10/10/10

Step-by-step explanation:

A function is a rule that generates one unique output value for every specified input value. Therefore, the fourth statement is true.

In mathematics, the function is something that relates an input to an output. The output obtained will somehow be related to the input. We write the function as f(x). For example, consider f(x) = x². This means that the function f takes the value of x and squares this value. Like when we input x=4. Then the output is f(x)=4²=16.

So, here see the function assigns a single and a unique output value of 16 for the input value of 4. Therefore, option 4 is correct. That is function gives every input value a unique, separate output value.

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The associative property shows that the value of 8/9 / 7/5 x 3/16 is 5/24.

The associative property states that the way in which we group the numbers or variables in an arithmetic operation does not affect the result. In other words, we can change the grouping of the numbers or variables by using parentheses without changing the outcome of the calculation.

In this problem, we can use the associative property to group the fractions in a way that makes it easier to simplify.

= 8/9 ÷ 7/5 x 3/16

= (8/9) ÷ (7/5) x 3/16

= 8/9 x 5/7 x 3/16

= 8/9 x 15/112

= 120/1008

= 5 / 24

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