KEY QUESTIONS Lesson 15 1. Application [4 marks] Another model for a bouncing ball is h(t)=4sin? (86t) where his height measured in metres and t is time in seconds. When is the first time the ball at a height of 2 m?

There are different types of interest rates that borrowers should be aware of: 1. Fixed Interest Rates, 2. Variable/Adjustable Interest Rates, 3. Prime Interest Rates, 4. Annual Percentage Rate (APR)

Interest rates refer to the percentage charged by lenders on borrowed funds, which borrowers must pay in addition to the principal amount. They represent the cost of borrowing money and play a significant role in determining the affordability and overall cost of loans. Higher interest rates imply higher borrowing costs for borrowers, while lower interest rates can make borrowing more affordable.

Interest rates can vary depending on several factors, including the type of loan, the borrower's creditworthiness, prevailing market conditions, and central bank policies. There are different types of interest rates that borrowers should be aware of:

1. Fixed Interest Rates: These rates remain constant throughout the loan term, providing borrowers with predictable monthly payments. Fixed rates are commonly used for mortgages and long-term loans, offering stability and protection against potential rate increases.

2. Variable/Adjustable Interest Rates: Also known as adjustable rates, these rates can change over time based on an underlying benchmark rate, such as the prime rate or the London Interbank Offered Rate (LIBOR). Variable rates are often lower initially but can fluctuate, leading to changes in monthly payments.

3. Prime Interest Rates: The prime rate is the interest rate offered to a bank's most creditworthy customers. It serves as a benchmark for many other interest rates, such as variable rate loans and credit cards. Borrowers with strong credit histories may qualify for loans with rates below the prime rate.

4. Annual Percentage Rate (APR): The APR represents the true cost of borrowing by factoring in both the interest rate and associated fees. It provides a comprehensive measure of the total cost of a loan and helps borrowers compare different loan offers.

Understanding the different types of interest rates and their implications is crucial for borrowers when considering loans. It is essential to evaluate the overall cost of borrowing, including interest rates, fees, and repayment terms, to make informed financial decisions. Borrowers should shop around, compare offers from different lenders, and consider their financial situation and long-term affordability before committing to a loan.

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1.1) The volume of the cube is 27 cubic centimeters. 1.2)the density of the ice cube is approximately 0.917 grams per cubic centimeter (g/cm³).

1.3) the mass of the water cube is 27 grams. 1.4) the weight of the water and the ice would be the same under the same conditions. 1.5)In simpler terms, ice floats on water because it is lighter (less dense) than the water, allowing it to displace an amount of water equal to its weight and float on the surface.

1.1) The volume of the cube can be calculated using the equation: Volume = width x height x length. In this case, the cube measures 3 cm wide, 3 cm tall, and 3 cm long, so the volume is:

Volume = 3 cm x 3 cm x 3 cm = 27 cm³.

Therefore, the volume of the cube is 27 cubic centimeters.

1.2) Density is defined as mass divided by volume. The mass of the ice cube is given as 24.76 grams, and we already determined the volume to be 27 cm³. Therefore, the density of the ice cube is:

Density = Mass / Volume = 24.76 g / 27 cm³ ≈ 0.917 g/cm³.

Therefore, the density of the ice cube is approximately 0.917 grams per cubic centimeter (g/cm³).

1.3) The volume of the water cube is the same as the ice cube, which is 27 cm³. Given the density of liquid water as 1.00 g/cm³, we can calculate the mass of the water cube using the equation:

Mass = Density x Volume = 1.00 g/cm³ x 27 cm³ = 27 grams.

Therefore, the mass of the water cube is 27 grams.

1.4) The weight of an object depends on both its mass and the acceleration due to gravity. Since the volume of the water cube and the ice cube is the same (27 cm³), and the mass of the water cube (27 grams) is equal to the mass of the ice cube (24.76 grams), their weights would also be equal when measured in the same gravitational field.

Therefore, the weight of the water and the ice would be the same under the same conditions.

1.5) Ice floats on water because it is less dense than liquid water. The density of ice is lower than the density of water because the water molecules in the solid ice are arranged in a specific lattice structure with open spaces. This arrangement causes ice to have a lower density compared to liquid water, where the molecules are closer together.

When ice is placed in water, the denser water molecules exert an upward buoyant force on the less dense ice, causing it to float. The buoyant force is the result of the pressure difference between the top and bottom surfaces of the submerged object.

In simpler terms, ice floats on water because it is lighter (less dense) than the water, allowing it to displace an amount of water equal to its weight and float on the surface.

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The measure of angle R is : 41.5°

Trigonometric ratios are ratios of the sides of a right-angled triangle.

They are used to relate the sides of a triangle with their angles.

Examples are cosine, tangent, sine etc.

Analysis:

length r is facing angle R, length s is facing angle S, length q is facing angle Q.

in order to find angle R we need to find the length of q first.

using cosine rule,

\(q^{2}\) = \(s^{2}\) + \(r^{2}\) -2srcosQ

\(q^{2}\) = \((30.6)^{2}\) + \((24.4)^{2}\) -2 x 30.6 x 24.4cos15

\(q^{2}\) = 936.6 + 595.36 -1493.3(0.9659)

q = 9.5

using sine rule,

\(\frac{24.4}{sin R}\) = \(\frac{9.5}{sin 15}\)

9.5 sinR = 24.4sin15

sin R = 24.4(0.2588)

sin R = 6.31

R = arc(sin) 6.31 = 41.5°

In conclusion, the measure of angle R is 41.5°

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Answer: (2x+6y)^2 = (2x+6y)(2x+6y) = 4x^2 +12xy + 12yx + 36y^2 = 4x^2 + 24xy + 36y^2

So the result of (2x+6y)^2 is 4x^2 + 24xy + 36y^2

Based on the time it took the patient to recover and the mean and standard deviation, the z-score is 1.5.

The z-score for this patient who takes 7 days to recover can be found as:

= (Time taken to recover - Mean) / Standard deviation

Solving for the z-score gives:

= (7 - 5.2) / 1.2

= 1.8 / 1.2

= 1.5

The first part of the question is:

The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.2 days and a standard deviation of 1.2 days.

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The transfer function for the given ODE is H(s) = 5 / (63s + 68). The transfer function relates the input function F(s) to the output function X(s) in the Laplace domain.

To determine the transfer function for the given ordinary differential equation (ODE), we need to apply the Laplace transform to both sides of the equation. The Laplace transform of a function f(t) is denoted as F(s) and is defined as:

F(s) = L[f(t)] = ∫[0 to ∞] e^(-st) f(t) dt

Applying the Laplace transform to the given ODE, we have:

38s + 30sX(s) + 63s^2X(s) = 5F(s)

Rearranging the equation and factoring out X(s), we get:

X(s) = 5F(s) / (38s + 30s + 63s^2)

Simplifying further:

X(s) = 5F(s) / (63s^2 + 68s)

Dividing the numerator and denominator by s, we obtain:

X(s) = 5F(s) / (63s + 68)

Thus, the transfer function for the given ODE is:

H(s) = X(s) / F(s) = 5 / (63s + 68)

Therefore, the transfer function for the given ODE is H(s) = 5 / (63s + 68). The transfer function relates the input function F(s) to the output function X(s) in the Laplace domain.

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The measure of angle fge is 53.13°.

The equation used to find the measure of angle fge is the Law of Cosines. The Law of Cosines states that for any triangle, the square of the length of any side is equal to the sum of the squares of the other two sides, minus twice the product of those sides and the cosine of the angle opposite the side.

Using the Law of Cosines, we can solve for the measure of angle fge as follows:

\(fge = cos-1((a^2 + b^2 - c^2) / 2ab)\)

For example, if the length of side a is 3, the length of side b is 4, and the length of side c is 5, then the measure of angle fge can be calculated as follows:

fge =\(cos-1((3^2 + 4^2 - 5^2) / 2(3)(4))\)

fge = \(cos-1(1/6)\)

fge = 53.13°

Therefore, the measure of angle fge is 53.13°.

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The benefit of using a curve over a histogram is that, it is a useful tool for visualizing data distributions, curves provide a more detailed and flexible representation of the data and offer additional advantages in terms of resolution, flexibility, parametric modeling, and comparison.

Visualizing data is an essential aspect of understanding the underlying patterns and relationships. In data analysis, two popular ways of visualizing data are histograms and curves. While both of these visualization techniques serve the purpose of summarizing data, each of them has its unique benefits and limitations.

Resolution: One of the main benefits of using a curve over a histogram is that a curve provides a higher resolution representation of the data. A curve can capture finer details of the data distribution that might get lost in a histogram due to the binning process.

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The use of a "thumbs-up" gesture to symbolize the statement "good luck" is a common practice in many cultures. This gesture is typically used to express approval or agreement, but it can also be used to convey positive wishes, such as wishing someone good luck before an exam or a job interview.

It is believed that this gesture originated in ancient Rome, where it was used as a sign of approval in the gladiatorial arena. Today, the thumbs-up gesture has become a universal symbol of positivity and encouragement. So, if you want to wish someone good luck, a thumbs-up gesture is a great way to do it!

The use of a "thumbs-up" gesture to symbolize the statement "good luck" involves the following steps:

1. Extend your arm: To perform the thumbs-up gesture, first extend your arm in the direction of the person you want to wish good luck.
2. Make a fist: Close your fingers into a fist, with your thumb pointing upwards.
3. Show your thumb: Ensure that the thumb is clearly visible and pointing upwards. This represents the "thumbs-up" gesture.
4. Make eye contact: Look at the person you're wishing good luck to, so they know the gesture is directed at them.

By performing the thumbs-up gesture, you're using a universally recognized symbol to convey your positive sentiment and wish someone good luck in their endeavors.

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Answer:

16x +2y +9

Step-by-step explanation:

I hope this helps. Just combine the numbers with the same Variable/letter. That's why only 7x and 9x were combined together

The structure with a formula of C₅H₁₃N, which has an integration of 9H, 3H, 1H, is pentylamine (C₅H₁₃N).

Pentylamine (C₅H₁₃N) is a primary amine with five carbon atoms (pentyl group) attached to a nitrogen atom. The molecular formula indicates that it contains 5 carbon atoms, 13 hydrogen atoms, and 1 nitrogen atom.

To determine the integration values, we count the number of chemically equivalent hydrogen atoms in the molecule. In pentylamine, there are three types of hydrogen atoms:

1. The amine group (-NH₂) has 2 hydrogen atoms attached to the nitrogen atom. These two hydrogens are chemically equivalent and are represented by the integration value of 2H.

2. The four carbon atoms directly bonded to the nitrogen atom each have three hydrogen atoms bonded to them. These twelve hydrogens are also chemically equivalent, resulting in the integration value of 12H.

3. The fifth carbon atom (end of the pentyl chain) has only one hydrogen atom bonded to it, which is represented by the integration value of 1H.

Therefore, the integration values of 9H, 3H, and 1H correspond to the three types of hydrogen atoms in the pentylamine molecule.

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Step-by-step explanation:

To estimate the difference between 5 1/2 and 3 5/9, we can first round the fractions to the nearest whole number or simpler fractions. In this case, we can round 1/2 to 1/2 and 5/9 to 1/2 as well. Now, we have:

5 1/2 − 3 1/2

Subtracting the whole numbers, we get:

5−3=2

Subtracting the fractions, we get:

1/2 − 1/2 = 0

So, the estimated difference is 2, which falls between 2 and 2 1/2. Therefore, Elise chose option D as the correct answer.

Answer:

The original price was $875.

Step-by-step explanation:

a = p * w

630 = 72% * w

630 / 0.72 = $875

$875 should be the original price, I hope this helps!

Answer:

see my explantion!!

Step-by-step explanation:

8000*0.05=400

400*18=7200$

Maximizing the sum or mean difference between the residuals squared, do not accurately describe the goal of the least squares regression line.

The least squares regression line minimizes the sum of the residuals squared. This means that it aims to minimize the difference between the observed values and the predicted values squared.

This is done by finding the line that has the smallest overall distance from the data points.

The other options, such as maximizing the sum or mean difference between the residuals squared, do not accurately describe the goal of the least squares regression line.

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The least squares regression line minimizes the sum of the residuals squared.

In regression analysis, the least squares regression line is a line that best fits a set of data points by minimizing the sum of the squared differences between the observed values and the predicted values.

The residuals are the differences between the observed values and the predicted values, and squaring them ensures that negative and positive differences are both considered.

By minimizing the sum of the squared residuals, the least squares regression line finds the line that is closest to all the data points.

For example, let's say we have a dataset of 10 data points.

The least squares regression line is determined by finding the line that minimizes the sum of the squared differences between the observed values and the predicted values for each data point.

This means that the line is positioned in a way that it is as close as possible to all the data points, minimizing the overall error.

By minimizing the sum of the residuals squared, the least squares regression line provides the best fit to the data and can be used to make predictions or analyze relationships between variables.

It is widely used in various fields, including statistics, economics, and social sciences.

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Answer:

x+4

.....................

Answer

\( \boxed{x + 4}\)

Option C is the correct option

Step by step explanation

Let's find the expression which represents the length of the box:

\( \mathsf{length \times width \times height \: of \: prism \: = \: volume \: of \: prism}\)

\( \mathsf{lengh \times \: (x - 1) \times (x + 8) = {x}^{3} + 11 {x}^{2} + 20x - 32}\)

\( \mathsf{length = \frac{ {x}^{3} + 11 {x}^{2} + 20x - 32 }{(x - 1)(x + 8)} }\)

Write 11x² as a sum

\( \mathsf{ = \frac{ {x}^{3} - {x}^{2} + 12 {x}^{2} + 20x - 32 }{(x - 1)(x + 8)} }\)

Write 20x as a sum

\( \mathsf{ = \frac{ {x}^{3} - {x}^{2} + 12 {x}^{2} - 12x + 32x - 32 }{(x - 1)(x + 8)}}\)

Factor out x² from the expression

\( \mathsf{ = \frac{ {x}^{2}(x - 1) + 12 {x}^{2} - 12x + 32x - 32 }{(x - 1)(x + 8)} }\)

Factor out 12 from the expression

\( \mathsf{ = \frac{ {x}^{2}(x - 1) + 12x(x - 1) + 32x - 32 }{(x - 1)(x + 8)} }\)

Factor out 32 from the expression

\( \mathsf{ = \frac{ {x}^{2}(x - 1) + 12x(x - 1) + 32(x - 1) }{(x - 1)(x + 8)} }\)

Factor out x+1 from the expression

\( \mathsf{ = \frac{(x - 1)( {x}^{2} + 12x + 32) }{(x - 1)(x + 8)} } \)

Factor out 12x as a sum

\( \mathsf{ = \frac{(x - 1)( {x}^{2} + 8x + 4x + 32) }{(x - 1)(x + 8)} }\)

Reduce the fraction with x-1

\( \mathsf{ = \frac{ {x}^{2} + 8x + 4x + 32 }{(x + 8)} }\)

Factor out x from the expression

\( \mathsf{ = \frac{x(x + 8) + 4x + 32}{(x + 8)} }\)

Factor out 4 from the expression

\( \mathsf{ = \frac{x(x + 8) + 4(x + 8)}{x + 8} }\)

Factor out x+8 from the expression

\( \mathsf{ = \frac{(x + 8)(x + 4)}{x + 8} } \)

Reduce the fraction with x+8

\( \mathsf{ = x + 4}\)

hence, x+4 is the expression that represents the length of a box.

Hope I helped!

Best regards!

The transformation of the given image follows the pattern; Reflection across the x-axis, then translation by 2 units downwards, the reflection across the y-axis.

From the given first image, we see that it is in the first quadrant but by the second image, we see that it is now mirrored along the horizontal axis. Thus, we can say that it is reflected across the x-axis.

After reflection, across the x-axis, we see that it was translated down by 2 units.

The last transformation is that we see it is now in quadrant 3 which means it was now reflected across the y-axis.

The transformation of the given image follows the pattern; Reflection across the x-axis, then translation by 2 units downwards, the reflection across the y-axis.

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Answer:

speed and distance covered

Step-by-step explanation:

when the distance is a constant, the time is related inversely.

Answer:

15

Step-by-step explanation:

Answer:  

Miguel babysits for 3 hours and earns $15. Which represents the unit rate?

Option  D

(D) $5 per hour

Step-by-step explanation:

$5.00 Per Hour Represents unit rate,  15 / 3 = 5

Answer:

480 and 481

Step-by-step explanation:

2 conservative

let's say the first number is x

the second number is x+1

so x + (x+1) = 961

2x + 1 = 961

2x = 960

x = 480

the first number is x , so it equals 480

the second number is x+1, so it equals 481

The sum of Peter, Richard, and John's ages is 191.

We know that the ratio of Peter's age to Richard's age is = 5/8 --(i)

The ratio of John's age to Peter's age is = 7/12 --(ii)

Similarly, the ratio of John's age to Richard's age is = (5*7)/(8*12)= 35/96 -(iii)

Using the value of (i),(ii),(iii), we get -

Age of Peter = (5*12)/(8*12) = 60/96

= 60 years of age

Age of John = (7*5)/(12*5) = 35/60

=35 years of age

From the value of (iii), since no one is over 100 years of age, the age of Richard= 96 years of age

Hence the sum of their ages is = 96+35+60

= 191

Therefore, we know that the sum of their ages is 191.

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Answer:

35. there is .25 chance. there are 12 tiles 3 are green so that is 1/4 of the tiles.

36. 1.5 inches for each spring. divide 450 by 300

Answer: \(e=-50|t-40|+2000\)

The cable car’s elevation will be 750 feet after 15 minutes or 65 minutes.

Step-by-step explanation:

Given: A cable car begins its trip by moving up a hill. As it moves up, it gains elevation at a constant rate of 50 feet/minute until it reaches the peak at 2,000 feet.

Then, total time taken to reach the peak = (Distance) ÷ (speed)

= (2,000 feet) ÷ ( 50 feet/minute)

= 40 minutes

Then, as the car moves down to the hill’s base, its elevation drops at the same rate.

The equation that models the cable car’s elevation, e, after t minutes is

e= (constant rate)|t- time to reach peak |+ Peak's height

\(e=-50|t-40|+2000\)

When the cable car’s elevation will be 750 feet after minutes, then we have

\(750=-50|t-40|+2000\\\\\Rightarrow\ -50|t-40|=750-2000\\\\\Rightarrow\ -50|t-40|=1250\\\\\Rightarrow|t-40|=-\dfrac{1250}{50}\\\\\Rightarrow|t-40|=-25\\\\\Rightarrow t-40=-25\text{ or }t-40=25\\\\\Rightarrow t=-25+40\text{ or }t=25+40\\\\\Rightarrow t=15\text{ or }t=65\)

Time cannot be negative, so the cable car’s elevation will be 750 feet after 15 minutes or 65 minutes.

Question 3: True

Question 4: False

Question 5: True

If the derivative of a function, f'(x), is positive for values of x less than c and negative for values of x greater than c, then it indicates a change in the slope of the function. This change from positive slope to negative slope suggests that the function has a maximum value at x = c.

This is because the function is increasing before x = c and decreasing after x = c, indicating a peak or maximum at x = c.

Question 3: If f'(c) < 0 then f(x) is decreasing and the graph of f(x) is concave down when x = c.

True

When the derivative of a function, f'(x), is negative at a point c, it indicates that the function is decreasing at that point. Additionally, if the second derivative, f''(x), exists and is negative at x = c, it implies that the graph of f(x) is concave down at that point.

Question 4: A local extreme point of a polynomial function f(x) can only occur when f'(x) = 0.

False

A local extreme point of a polynomial function can occur when f'(x) = 0, but it is not the only condition. A local extreme point can also occur when f'(x) does not exist (such as at a sharp corner or cusp) or when f'(x) is undefined. Therefore, f'(x) being equal to zero is not the sole requirement for a local extreme point to exist.

Question 5: If f'(x) > 0 when x < c and f'(x) < 0 when x > c, then f(x) has a maximum value when x = c.

True

If the derivative of a function, f'(x), is positive for values of x less than c and negative for values of x greater than c, then it indicates a change in the slope of the function. This change from positive slope to negative slope suggests that the function has a maximum value at x = c. This is because the function is increasing before x = c and decreasing after x = c, indicating a peak or maximum at x = c.

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Answer:

yes, y=2x-3 is linear

2x - y = 3  Standard Form

Step-by-step explanation:

since both the 'x' and 'y' variables have one as their exponents then this represents a linear equation

standard form: Ax + By = C

y = 2x - 3

subtract 2x from each side to get:

-2x + y = -3

divide each term by -1 to get (this is done so the leading coefficient is not negative)

2x - y = 3  Standard Form

Answer:

Yes, it's a linear equation. The standard form is 2x - y = 3.

Step-by-step explanation:

The equation y = 2x - 3 is in slope-intercept form. Slope-intercept form applies to linear equations/functions.

To convert slope-intercept into standard:

1. Subtract 2x from both sides.

2. Multiply everything by -1.

Have a lovely rest of your day/night, and good luck with your assignments! ♡

 ~ ren ⚘

Answer:

only ingrid

Step-by-step explanation: