A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 100cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds.
Determine the spring constant k.
k = ? Newtons / meter
Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y,y′,y′′,t.)
Differential equation: ?
Initial conditions: y(0) = ? and y′(0) = ?
Solve the initial value problem for y(t).
y(t) = ?
Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 ≤ t < [infinity]. If there is no such maximum, enter NONE.
maximum excursion = ? meters

The values of unknown angles in the given figures are:

(a) a° = 117°, b° = 117°, c° = 63°, d° = 78°

(b) e° = 31°, f° = 66°

(c) g° = 69°, h° = 83°

(d) i° = 45°, j° = 60°

Angles are formed when two rays join at a common point. The common point can also be called as a node or vertex, and the two rays are called angle arms. Angles are usually represented by the "∠" symbol. The word angle is derived from the Latin "Angulus". Learn more about lines and angles here.  Angle types are based on angle values ​​in degrees. You can also express angles in radians, or pi (π). 180° is equivalent to pi radians.  

For the given figures:

a° = 117° [ Vertically opposite angles]

b° = 117° [Corresponding angle]

c° = 180° - 117° [Linear pair]

c° = 63°

d° = 78° [Corresponding angle]

e° = 31° [Alternate interior angles]

f° = 66° [Alternate interior angles]

g° = 69° [Corresponding angle]

h° = 83° [Alternate interior angles]

j° = 60° [Alternate interior angles]

i° = 180 - (75 + 60) [Angle sum property of triangle]

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Answer:

Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer

The best deal for Jack and Jill would be for Jill to offer Jack $10, and for Jack to accept this offer. This would give Jack $9.09 in the future after discounting, and it would give Jill $10 in the future after discounting.

Jack will accept anything positive.

Jill is indifferent between $20 in one year and $16 today (she discounts the future at 20%).

Jack is indifferent between $4 in one year and $3.60 today (he discounts the future at 10%). Note that $16.40 is preferred by Jill to $16 in one year.

Discount refers to the process of reducing the value of future benefits or costs, based on the assumption that people generally prefer immediate benefits to delayed ones. This means that the further into the future a benefit or cost occurs, the less valuable it is considered to be in the present.

Discounting is commonly used in economics, finance, and environmental analysis, where it plays a significant role in determining the present value of future cash flows, investments, and environmental damages. The concept is also used in decision-making, as individuals and organizations often make choices based on their perception of the present value of future outcomes.

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The pοint that satisfies bοth inequalities is the pοint inside this triangular regiοn.

An inequality is a mathematical statement that cοmpares twο values οr expressiοns and indicates whether they are equal οr nοt, οr which οne is greater οr smaller.

Since the shading is nοt included, we will need tο use the lines themselves tο determine the cοrrect regiοn οf the cοοrdinate plane.

The first inequality y > (3/2)x - 5 has a slοpe οf 3/2 and a y-intercept οf -5. This means the line will have a pοsitive slοpe and will be lοcated belοw the pοint (0,-5).

The secοnd inequality y < (-1/6)x - 6 has a negative slοpe οf -1/6 and a y-intercept οf -6. This means the line will have a negative slοpe and will be lοcated abοve the pοint (0,-6).

Tο find the pοint that satisfies BOTH inequalities, we need tο lοοk fοr the regiοn οf the cοοrdinate plane that is belοw the line y = (3/2)x - 5 AND abοve the line y = (-1/6)x - 6. This regiοn is the triangular-shaped area that is bοunded by the twο lines and the x-axis.

The pοint that satisfies bοth inequalities is the pοint inside this triangular regiοn.

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Answer:

52.17%

Step-by-step explanation

[tex]\frac{12000}{23000}[/tex] ×100%

[tex]\frac{1200}{23}[/tex]

52.17

A manufacturer of compact fluorescent light bulbs advertises have a standard deviation of 1,000 hours so the values are:

A normal distribution with,

μ = 9000

σ = 1000

a) The standardized score is the value x decreased by the mean and then divided by the standard deviation.

x = 105000 - 9000 / 1000 ≈ 1.50

Determine the corresponding probability using the normal probability table in appendix,

P(X>10500) = P(Z>1.50) = 1 - P(Z<1.50)

= 1 - 0.9332 = 0.0668.

b) n = 15

The sampling distribution of the mean weight is approximately normal, because the population distribution is approximately normal.

The sampling distribution of the sample mean has mean μ and standard deviation σ/√n

μ = 1000/√15 = 258.19

c) The sampling distribution of the sample mean has mean μ and standard deviation σ/√n

The z-value is the sample mean decreased by the population mean, divided by the standard deviation:

z = x-u/σ/√n = 10500-9000/1000√15 = 5.81

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Answer:

The ratio of green legos to missing legos is:

42 : 3

Simplifying this ratio by dividing both sides by 3, we get:

14 : 1

Therefore, for every one missing green lego, there are 14 green legos.

Answer:

-8 < k

Step-by-step explanation:

4 < 4(k+3)

4 < 4k + 12

-8 < k

Answer:

logistic growth

Step-by-step explanation:

In logistic growth, a population's per capita growth rate gets smaller and smaller as population size approaches a maximum imposed by limited resources in the environment, known as the carrying capacity ( K).

Answer: To fill out a table of equivalent ratios for Blake's lemonade recipe, we can use the ratio of lemons to cups of lemonade:

Lemons             Cups of Lemonade

3                        4

6                        8

9                        12

12                      16

15                      20

18                      24

To plot the points on a coordinate axes, we can use the Lemons as the x-coordinate and the Cups of Lemonade as the y-coordinate. The points would lie on a line that passes through the origin (0,0) and the point (12,16).

         |       *

 Cups of |    *     *

Lemonade  | *           *

         |________________

            Lemons

              0    12

The line represents the proportional relationship between the number of lemons and the amount of lemonade produced. As the number of lemons increases, the amount of lemonade produced increases proportionally.

Step-by-step explanation:

The value of n, which represents the number of sides in polygon B, is 160.

Geometry is a branch of mathematics that deals with the study of shapes, sizes, positions, and properties of objects in space.

The given diagram shows parts of three regular polygons, labeled as A, B, and C, meeting at a point. Angles within the polygons are labeled with their respective measures in degrees.

According to the diagram:

Since polygons A, B, and C are regular polygons, all their interior angles are equal. Let's denote the measure of each interior angle in polygon B as n°.

Now, we can set up an equation based on the angle measures in the diagram:

3x + 8x + 7x = 360

Simplifying the equation:

18x = 360

Dividing both sides by 18 to solve for x:

x = 360 / 18

x = 20

Now, substituting the value of x back into the equation for n° in polygon B:

8x = 8 * 20 = 160

Hence, the value of n, which represents the number of sides in polygon B, is 160.

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Complete question:

Answer:

4,320 times

Step-by-step explanation:

12 x 6 = 72 beats per minute

72 x 60 = 4320 beats in 60 minutes

Answer: 840 in 30 minuets

Step-by-step explanation:

Answer:

Anita is 11 years old and sarun 55 years old( not so sure about this answer... what do you think)

Step-by-step explanation:

let Anita's age be x

and sarun's age be 3x

if, x-5 = 3x+7

3x+7 = 5(x-5)

3x+7=5x-25

32=2x

x=16

their ages 3 years ago,

Anita= 16-5=11yrs

sarun= 3*16+7 = 55yrs

11.87 m² metal sheet would be needed to make the base area of tank.

Volume of cylinder, determines how much material it can carry, is determined by the cylinder's volume. A cylinder is a three-dimensional structure having two parallel, identical bases that are congruent.

It is given that capacity of a closed cylindrical vessel of height 2 m is 3080 liters

Let us assume that Radius of cylinder = r

Then Volume of cylinder = π ×r² ×h

= 2π ×r²× m³

1 m³ = 1000 liters

= 2000 π r²  liters

Volume of tank = Capacity

2000 π r² = 3080

=> 2000 × (22/7) × r² = 3080

=> r² = 49/100

=> r = 7/10 m

=> r = 0.7 m

Base Area of tank = TSA = 2πrh + 2πr²

= 2×(22/7)(0.7)×2 + 2×(22/7)×(0.7)²

= 3.0772 +8.792

= 111.87 m²

Hence, 11.87 m² metal sheet would be needed to make it.

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By combinations, we get that there are 20,020 possible 6-topping pizzas.

Given that a pizza parlor offers a choice of 16 different toppings, we need to find the number of 6-topping pizzas that are possible. No double-orders of toppings are allowed. What is the number of 6-topping pizzas that are possible?

The formula for combinations is as follows:

ncr = (n!)/(r! * (n - r)!) where n = 16 (total number of toppings available)

r = 6 (number of toppings required for a pizza)

ncr = (16!)/(6! * (16 - 6)!)

⇒ ncr = (16!)/(6! * 10!)

⇒ ncr = (16 * 15 * 14 * 13 * 12 * 11)/(6 * 5 * 4 * 3 * 2 * 1)

⇒ ncr = (16 * 3 * 5 * 7 * 13 * 11)/(3 * 2 * 2)

⇒ ncr = 20,021

In this situation, the order of the toppings does not matter. Therefore, we cannot have a repeat of the same toppings, as mentioned. Hence the answer to the question is no. Thus, there are 20,020 possible 6-topping pizzas.

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The minimum value of the sum of squares is 3 when all three numbers are equal to 1.

Given the condition that the product of three positive integers x, y, and z is 1 and the sum of the squares is a minimum, we have to find out the three numbers that satisfy this condition.The three numbers that satisfy the condition arex = 1, y = 1, and z = 1 because the product of any number and 1 is the number itself, and the sum of the squares of all the numbers is also 1. Thus, the answer is (1,1,1).It can be proved mathematically that the answer (1, 1, 1) is correct:Product of three positive integers is 1 if and only if all three numbers are equal to 1.If three positive integers a, b, and c have a fixed sum, then the sum of the squares of two of the integers is the minimum when they are equal to each other and their sum is equal to half of the fixed sum.Thus, the sum of the squares of the three numbers is 1 + 1 + 1 = 3, and the minimum value of the sum of squares is 3 when all three numbers are equal to 1.

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The cost per ounce of the juice is $0.0.9 using unitary method.

The unitary approach is a strategy for problem-solving that involves first determining the value of a single unit, then multiplying that value to determine the required value. The unitary method, in its most basic form, is used to calculate the value of a single unit from a specified multiple. The unitary technique can be used to complete it. Additionally, after determining the value of a single unit, we can increase that value by the number of units needed to determine the value of the additional units. What is division?One of the four fundamental arithmetic processes, or how numbers are combined to create new numbers, is division. The other processes are addition, subtraction, and multiplication. The reverse of multiplying is division.

In this question,

64-ounce container of juice costs $5.76. I

ounce of juice costs = 5.76/64= $0.09

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This means that, by lowering the confidence level, it is easier to reach the desired level of statistical significance and thus easier to conclude that the number of days above 90 degrees has increased.

F) Lowering the degrees of freedom would make it easier to conclude that there are more days above 90 degrees in 2018 compared to 1948 globally. The reason for this is that reducing the degrees of freedom makes it easier to achieve the desired statistical significance. By reducing the degrees of freedom, you are essentially decreasing the number of observations needed in order to reach the desired confidence level. This makes it easier to conclude that the number of days above 90 degrees has increased since 1948, as it requires fewer observations to reach the desired significance level.

G) Lowering the confidence level would make it easier to conclude that there are more days above 90 degrees in 2018 compared to 1948 globally. This is because a lower confidence level requires less data in order to achieve the desired statistical significance. By reducing the confidence level, fewer observations are needed in order to prove that the number of days above 90 degrees has increased since 1948. This means that, by lowering the confidence level, it is easier to reach the desired level of statistical significance and thus easier to conclude that the number of days above 90 degrees has increased.

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A fair coin is flipped 40 times. The probability of getting a head when the coin is tossed is 0.5. If the same coin is flipped 40 times, the probability will remain 0.5. That is, there are equal chances of getting heads and tails when a fair coin is flipped.

Based on the confidence interval, if the actual proportion of heads falls within the range of the confidence interval, it can be said that the coin used was fair. If the actual proportion is outside the confidence interval, it may be an indication that the coin was not fair.

The level of confidence is typically 95% or 99%. If a confidence interval is constructed for the proportion of heads based on a sample of 40 flips and the interval includes the expected proportion of 50%, it can be said that the coin used was fair. If the interval does not include 50%, there is evidence that the coin may not be fair.

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The answer is 45 toy soldiers which can be solved by applying the fundamentals of algebra.

Algebra is a branch of mathematics which uses symbols to represent numbers and variables in equations and formulas to solve problems. Algebra is used to understand the relationships between different values, to find unknown values, and to solve equations.

To represent the two scenarios provided in the question, we must first build up two equations.

If Leo arranges the toy troops in the first scenario four by four, then the equation is 4x = 27, where x is the total number of toy soldiers.

If Leo arranges the toy troops seven times out of seven in the second scenario, the equation is 7x = 54, where x is the total number of toy soldiers.

The equations must then be solved in order to determine the value of x in each equation.

We multiply both sides of the equation by 4 to get x = 6.75 for the first equation.

To get x = 7.71428 we multiply both sides of the second equation by 7.

We settle on 6, the lowest whole number between 6.75 and 7.7142, because there must be an even number of toy soldiers. In the first situation, Leo has a total of 24 toy soldiers, while in the second, he has 42, or 4 x 6 and 7 x 6, respectively.

Since the question specifies that there should be between 27 and 54 toy soldiers, we choose the higher value of 42 to arrive at the correct response, which is 45 toy soldiers.

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The R9,000 loan offer can be analyzed as follows;

A loan is a financial agreement that is made between a lender and a borrower, such that the borrower accepts a certain amount of money provided by the lender, which is to be paid back after a specified period of time, with an interest.

The total amount which has to be paid as repayment for the loan can be obtained by finding the product of the monthly installment and the total number of months to repay the loan:

Total amount Theo pays = R318.92 × 48 = R15308.16

The reasons banks and other financial institutions offer loans to people that did not apply for loans includes;

Theo's investment options can be analyzed as follows;

Option 1: Investment with simple interest calculation

Interest earned = R9,000 × 8% × 4 years = R2,880

The total value of the investment after 4 years = R9,000 + R2,880 = R11,880

Option 2: Investment with compound interest calculation

The total value of the investment after 4 years = R9,000 × (1 + 7%)⁴ = R11,797.16409

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Answer:

-3.375

Step-by-step explanation:

the point of a geometric sequence is that every new term is created by multiplying the previous term by a constant factor that is the same across the whole sequence.

that factor is called the common ratio.

so, 32 was multiplied by the common ratio, and we get -108 as result.

32 × cr = -108

cr = -108/32 = -27/8 = -3.375 = -3 3/8

The sale price including tax is $ 2390.625.

Given that,

The price of the snow machine= $ 1800

Discount percentage= 25%

Sales tax rate = 6.25%

Hence, Sale price = Price + Discount percentage of price

                              = 1800 + 25/100 × 1800

                              = 1800 + 450

                              = 2250.

Sale price including tax = Sale price + Tax rate

                                       = 2250 + 6.25/100

                                       = 2250 + 140.625

                                       = $ 2390.625.

Hence, the sale price including tax is $ 2390.625.

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As a result of demonstrating that two pairs of corresponding sides and their included angles are congruent in both triangles

A triangle is a closed, double-symmetrical object composed of three line segments known as sides that intersect at three points known as vertices. Triangles are distinguished by their sides and angles. Triangles can be equilateral (all factions equal), isosceles, or scalene based on their sides. Triangles are classified as acute (all angles are less than 90 degrees), okay (one angle is equal to 90 degrees), or orbicular (all angles are greater than 90 degrees) (all angles greater than 90 degrees). The region of a triangle can be calculated using the formula A = (1/2)bh, where an is the neighbourhood, b is the triangle's base, and h is the triangle's height.

Triangle ABC with AB, BC, and AC sides.

Triangle DEF with DE, EF, and DF sides.

Angle ABC and angle DEF are vertical angles, which means they are congruent, as shown in the diagram. As a result, angle ABC = angle DEF.

We can see from the diagram that BC and EF are both vertical, so they must be congruent because they are opposite sides of a parallelogram. As a result, BC = EF.

Because they are opposite sides of a parallelogram, they must be congruent. As a result, AC = DF.

As a result of demonstrating that two pairs of corresponding sides and their included angles are congruent in both triangles, we can conclude that Triangle ABC and Triangle DEF are congruent using the Side-Angle-Side (SAS) criterion.

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Thomas and Sadio have a 15:8 savings ratio that will be ready in five weeks.

In mathematics, a ratio shows how frequently one number appears in another.

For instance, the proportion of oranges to lemons in a fruit plate would be eight to six if there were eight oranges and six lemons.

Oranges make up 8:14 of the total fruit, whereas lemons make up 6:8 of the total fruit.

So, the number of weeks would be:

Considering that Sadio savings = 15 + 4.50 times

Thomas = x4

Where the number of weeks is x.

Sadio : Thomas  =  15 : 8

15 + 4.50x : 4x =  15 : 8

(15+ 4.50x ) / 4x =  15 / 8

Now, perform cross multiplication as follows:

8 (15+ 4.50x) = 15 (4x)

120 + 36x = 60x

60x - 36x = 120

24x = 120

x = 120 / 24 = 5 weeks

Therefore, Thomas and Sadio have a 15:8 savings ratio that will be ready in five weeks.

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Correct question:

Sadio starts with savings of £15. Thomas starts with no savings. Each week from now, Sadio will save £4.50 and Thomas will save £4. In how many weeks will Sadio and Thomas have savings in the ratio 15:8 ?

According to the inscribed angle theorem the measure of angle Y is 43 degrees.

As the inscribed angle equals half of the central angle, the inscribed angle theorem is also known as the angle at the centre theorem. The centre angle is always the same since the endpoints are fixed, regardless of where it is on the same arc between the ends. The measure of the central angle is equal to double the measure of the inscribed angle occupied by the same arc, according to this theorem. OR. Half of a central angle that subtends the same arc is an inscribed angle. OR. The angle in a circle's centre is twice as large as any angle at its perimeter that the same arc occupies.

We know that, an inscribed angle is half of a central angle that subtends the same arc.

Here the given arc = 86 degrees.

The inscribed angle is half the arc of the circle thus,

angle Y = 1/2(86)

angle Y = 43 degrees.

Hence, according to the inscribed angle theorem the measure of angle Y is 43 degrees.

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Answer:

0.999

Step-by-step explanation:

If the driver wore a seat belt, that is the only column we have to look at.

The chance the driver survived is number survived / number total.

Plugging in, we get 412,042/412,493, or about 0.999.

Hope this helps!

The estimated probability is 0.9537 (rounded to four decimal places).Hence, the answer is 0.9537.

Question: Nine percent of men are color blind. Researchers need at least 60 men with this trait, so they randomly select 700 men. Estimate the probability that at least 60 color-blind men are in the sample. Use a TI-83, TI-83 plus, or TI-84 calculator to find the probability. Round your answer to four decimal places. Provide your answer below:

Solution:

Let the probability of the man having the trait of color blindness be p. Then, p = 0.09, and the sample size is n = 700.Mean and Variance of Binomial Distribution are:μ = np and σ² = np(1 − p)Here, p = 0.09, n = 700μ = np = 700(0.09) = 63Andσ² = np(1-p) = 700(0.09)(0.91) = 5.679Now we need to find the probability that at least 60 color-blind men are in the sample.i.e., P(X≥60)By using the Central Limit Theorem, we can approximate this binomial distribution as a normal distribution with mean μ = 63 and variance σ² = 5.679.P(X≥60) can be calculated using the following formula:P(Z≥((60-63)/√(5.679))) = P(Z≥(-1.684))Now we need to look at the z-table or use the calculator to find the value of P(Z≥(-1.684))P(Z≥(-1.684)) = 0.9537 (rounded to four decimal places)Therefore, the estimated probability is 0.9537 (rounded to four decimal places).Hence, the answer is 0.9537.

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The car travelled at 80km/h for 7 hours and travelled a distance of 560km whereas the ferry travelled at 16km/h for 20 hours and travelled a distance of 320km.

We know that the Anderson’s travelled for 880 km from St. Johns to Halifax via Argentia and that part of the trip was by car at 80 km/h and the rest by ferry at 16 km/h.

we need to find how much time did it take the Anderson’s to travel by car and by ferry

as part of the journey was done by car and the other by ferry

560km was travelled by car therefore now we need to find time travelled by car = distance = speed x time

we get time travelled by car = 7 hours

similarly we need to find the time take by ferry we have:

320km travelled by ferry at 16km/h we get the time travelled by ferry = 20hours

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The required time is mathematically given as

t-150 minutes.

This is further explained below.

We know that the volume of the prism is given by.

[tex]\text{Volume}=\text{(Base Area)}\times\text{height}[/tex]

[tex]\text{Base Area}=\dfrac{1}{2}\times(1.4+0.6)\times2[/tex]

[tex]\text{Base Area}=2\ \text{m}^2[/tex]

Now,

[tex]\text{Volume}=2\times1=2\ \text{m}^3[/tex]

Given that there is a twenty percent drop in the level of the pond during the first thirty minutes.

Because of this, the volume increases.

[tex]\text{Volume}=2\times0.8=1.6\ \text{m}^3[/tex]

Now, the rate of emptying the tank is,

[tex]=\dfrac{2-1.6}{30} =\dfrac{1}{75} \ \text{m}^3/ \text{min}[/tex]

In conclusion, We also know that,

$$

[tex]\text{time}=\dfrac{\text{volume}}{\text{rate}}[/tex]

[tex]\text{time}=\dfrac{2}{\frac{1}{75} } =150 \ \text{min}[/tex]

Therefore, Colin must wait for the pond to drain for a total of 150 minutes.

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