Find the area of the trapezoid. 1. 6 m
4 m
12 m
2. 10 in. 15 in. 6 in
3. 2 yd
8 yd
6 yd
4. H = 10 cm, b,
= 3 cm, by = 5 cm

The total surface area of the triangular prism is calculated to be equal to 608 square centimeters.

The triangular prism can be observed to be a large rectangle and two identical triangles, so we shall calculate for the total surface area as follows:

area of one triangle = 1/2 × 8 cm × 12 cm

area of one triangle = 48 cm²

area of the two triangle faces = 2(48) = 96 cm²

area of the large rectangle face = (10 + 12 + 12) cm × 16 cm = 512 cm²

Total surface area of prism = 96 cm² + 512 cm² = 608 cm²

Therefore, the total surface area of the triangular prism is calculated to be equal to 608 square centimeters.

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the points we plotted are (-6, -1/2), (-4, 1/2), (0, -1/5), (3, 1/2), and (6, 1/5).

What is function?

a function is a relationship or expression involving one or more variables.  It has a set of input and outputs.

To find the x-intercepts, we set the numerator of the function equal to zero:

[tex]2 / (x^2 + 3x - 10) = 0[/tex]

This is only true if the numerator is equal to zero, so:

2 = 0

This is not possible, so the function has no x-intercepts.

To find the vertical asymptotes, we look for values of x that make the denominator equal to zero:

[tex]x^2 + 3x - 10 = 0[/tex]

We can factor the quadratic as:

(x + 5)(x - 2) = 0

This means that the function has vertical asymptotes at x = -5 and x = 2.

Now, let's plot some points between and beyond each x-intercept and vertical asymptote:

When x = -6, f(x) = 2 / (-6)² + 3(-6) - 10 = -1/2

When x = -4, f(x) = 2 / (-4)² + 3(-4) - 10 = 1/2

When x = 0, f(x) = 2 / (0)² + 3(0) - 10 = -1/5

When x = 3, f(x) = 2 / (3)² + 3(3) - 10 = 1/2

When x = 4, f(x) = 2 / (4)² + 3(4) - 10 = -1/2

When x = 6, f(x) = 2 / (6)² + 3(6) - 10 = 1/5

Therefore, the points we plotted are (-6, -1/2), (-4, 1/2), (0, -1/5), (3, 1/2), and (6, 1/5).

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The total surface area of each figure are:

1)  2,557.3 yd²

2) 601.02 m₂

3) 3782.5 mm²

4)750 in²

1) The total surface area of a cylinder is:

2πrh + 2πr².

where:

r is radius

h is height

Thus:

TSA = 2π(11 * 26) + 2π(11)²

TSA = 2π(286) + 2π(121)

TSA = 2π(407)

TSA = 2,557.3 yd²

2) The total surface area is:

2(¹/₂ * 9 * 11.12) + (16 * 15) + (16 * 12) + (9 * 16)

= 25.02 + 240 + 192 + 144

= 601.02 m₂

3) The total surface area of a cylinder is:

2πrh + 2πr².

where:

r is radius

h is height

Thus:

TSA = 2π(14 * 29) + 2π(14)²

TSA = 2π(406) + 2π(196)

TSA = 2π(602)

TSA = 3782.5 mm²

4) The total surface area of the pyramid is:

TSA = (15 * 8) + (21 * 17) + (21 * 15) + (8 * 21)

TSA = 120 + 147 + 315 + 168

TSA = 750 in²

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Therefore , the solution of the given problem of equation comes out to be 10x - y = 51 is the point-slope form of the equation of the line passing through the points (6, -9) and (7, 1).

Variable words are frequently used in complex algorithms to show consistency between two arguments that are opposed to one another. Academic expressions called equations are used to show the equality of various academic figures. Take into account the details in the z + 7 proposals.

Here,

The equation for the line passing between the points (6, -9) and (7, 1) must be found in the point-slope form.

Using the equation m = (y2 - y1) / (x2 - x1), find the slope (m) of the line, where (x1, y1) and (x2, y2) are the coordinates of the two points.

=> (x1, y1) = (6, -9)

=> (x2, y2) = (7, 1)

=> m = (1 - (-9)) / (7 - 6)

=> m = 10 / 1 m = 10

Consequently, the line's slope is 10.

=> y - y1 = m(x - x1)

=> m = 10 (x1, y1) = (6, -9)

=> y - (-9) = 10(x - 6)

=> y + 9 = 10x - 60

=> 10x - y = 51

Therefore, 10x - y = 51 is the point-slope form of the equation of the line passing through the points (6, -9) and (7, 1).

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The distance between your location and the epicenter is just slightly larger than the maximum distance that the earthquake can be felt (76 miles), so you would be able to feel the earthquake.

A triangle is a polygon with three sides and three angles. The sum of the angles in a triangle is always 180 degrees. There are different types of triangles such as equilateral, isosceles, scalene, right-angled, obtuse-angled, and acute-angled triangles. Triangles are used in geometry and other fields of mathematics to solve problems related to areas, angles, and side lengths.

Here,

Yes, you feel the earthquake.

To see why, imagine drawing a circle around the epicenter with a radius of 76 miles. This circle represents the maximum distance that the earthquake can be felt. Then, draw a line from the epicenter to your location. This line represents the distance between you and the epicenter.

To determine whether you feel the earthquake, we need to calculate the distance between your location and the epicenter using the Pythagorean theorem:

distance = √(65² + 40²)

distance ≈ 76.06 miles

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The answer to this question is  x = 6

Answer: C. experimental study.

Experimental studies are used to establish cause-and-effect relationships between variables by manipulating one variable (independent variable) and observing the effect on another variable (dependent variable) while controlling for other potential factors. Correlational studies examine the relationship between two variables but do not establish causality, descriptive studies describe a phenomenon without manipulating variables, and meta-analysis is a statistical method that combines the results of multiple studies to provide an overall summary.

Step-by-step explanation:

Below is the correct matching of the questions to the right answers

The statistical measures listed in the previous question are often used to describe and analyze statistical variables.

For instance, the range of a data set is a measure of the spread of values in a variable, while the quartiles and interquartile range provide information about the distribution of the variable. Mean average deviation is a measure of the variability of the variable around its mean value.

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1. Mean average deviation - The average deviation of the data from the mean.

2. Range of a data set - The difference between the highest value and the lowest value in a numerical data set.

3. First quartile - The median in the upper half of the rank-ordered data.

4. Second quartile - The median value in the data set.

5. Third quartile - The median in the lower half of the rank-ordered data.

6. Interquartile range - he distance between the first and third quartiles of the data set.

The statistical measures are often used to describe and analyze statistical variables. For instance, the range of a data set is a measure of the spread of values in a variable, while the quartiles and interquartile range provide information about the distribution of the variable. Mean average deviation is a measure of the variability of the variable around its mean value.

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The unit for the coefficient k would be dollars per unit of length squared.

To find the units for the coefficient k in the formula C = klw, we need to analyze the units of each variable.

C represents the cost of the home, which is measured in dollars.

l represents the length of the home, which is measured in some units of length, such as feet or meters.

w represents the width of the home, which is also measured in some units of length, such as feet or meters.

Therefore, we can determine the units for k by analyzing how the units for C, l, and w combine in the formula C = klw.

Starting with the left-hand side of the equation, we know that C represents dollars.

On the right-hand side of the equation, we have k times l times w. To determine the units of k, we need to figure out what units would make the entire right-hand side of the equation equal to dollars.

Since we are multiplying three quantities (k, l, and w), the units of k must cancel out the units of length in l and w in order to leave us with units of dollars.

This means that the units of k must be dollars per unit of length squared. We can write this as:

k = $/length^2

So, the units for the coefficient k in the formula C = klw are dollars per unit of length squared.

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The length of chord XY is 5√2.

What is the length of the chord?

It is described as the line segment that connects any two points on the circle's circumference without going through the circle's center. As a result, the diameter is the chord of a particular circle that is the longest and goes through its center. In mathematics, determining the chord's length can be crucial at times.

Here, we have

Given: A circle with center O and radius 5 has a central angle XOY. X and Y are points on the circle. ​If the measure of arc XY is 90 degrees.

We have to find the length of chord XY.

∠XOY = 90°

OX = OY = 5

We draw a perpendicular from the center to chord XY bisect XY at D.

Now, since OD bisects ∠XOY

∠XOD = ∠YOD = 90°/2 = 45°

Now, in ΔXOD

sin45° = XD/OX

1/√2 = XD/5

5/√2  = XD...(1)

In ΔYOD

sin45° = YD/OY

5/√2  = YD...(2)

Adding (1) and (2), we get

XD + YD = 5/√2 + 5/√2

XY = 5√2

Hence, the length of chord XY is 5√2.

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The value of arc PQ is 110°

The circumference of a circle is the length of the 360 degree arc of that circle. Therefore the sum of the arc on a circumference of a circle is 360°

Therefore ,

8x-10+ 6x + 10x+10 = 360

24x -10+10 = 360

24x = 360

divide both sides by 24

x = 360/24

x = 15

therefore the value of x is 15

Arc PQ = 8x -10

substitute 15 for x

ArcPQ = 8×15-10

= 120-10

= 110°

therefore the value of arc PQ is 110°

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( 5 - 2t ) < - 4 is an inequalty that represents t, the number of hours past midnight, when the temperature was coler than -4 degrees celsius.

What is  linear equation?

An algebraic equation with simply a constant and a first- order( direct) element, similar as y = mx b, where m is the pitch and b is the y- intercept, is known as a linear equation.

                         The below is sometimes appertained to as a" direct equation of two variables," where y and x are the variables. Equations whose variables have a power of one are called direct equations. One illustration with only one variable is where layoff b = 0, where a and b are real values and x is the variable.

The midnight temperature is 5 °C and the temperature is decreasing at the rate of 2°C per hour.

If t is the hours past midnight then after t hours the temperature will be ( 5 - 2t ).

Now, if this temperature is colder than - 4° C, then the inequality can be written as   ( 5 - 2t ) < - 4.

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The inequality that represents t, the number of hours past midnight, when the temperature was cooler than -4 degrees celsius( 5 - 2t ) < - 4

What is  inequality?

The term "inequality" is used in mathematics to describe a relationship between two expressions or values that is not equal to one another. Inequality results from a lack of balance. When two quantities are equal, we use the symbol '=', and when they are not equal, we use the symbol. If two values are not equal, the first value can be greater than (>) or less than (), or greater than equal to () or less than equal to ().

The midnight temperature is 5 °C and the temperature is decreasing at the rate of 2°C per hour.

If t is the hours past midnight then after t hours the temperature will be

=>  ( 5 - 2t ).

Now, if this temperature is colder than - 4° C, then the inequality can be written as   ( 5 - 2t ) < - 4.

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A reasonable estimate of the probability that Patsy sells fewer than 2 feathers at the next festival is about 0.2485.

How to estimate the probability of given data?

To estimate the probability that Patsy sells fewer than a certain number of feathers at the next festival, we need to use the given data to calculate the mean and standard deviation of the number of feathers she has sold in the past festivals.

The mean, denoted by μ, is calculated by adding up all the number of feathers sold and dividing by the total number of festivals. So,

μ = (3 + 6 + 1 + 4 + 2 + 3 + 7 + 2) / 8 = 3.375

The standard deviation, denoted by σ, is a measure of how spread out the data is. It is calculated by first finding the variance, which is the average of the squared differences between each data point and the mean, and then taking the square root of the variance. So,

Variance = ((3-3.375)² + (6-3.375)²+ (1-3.375)² + (4-3.375)² + (2-3.375)² + (3-3.375)² + (7-3.375)² + (2-3.375)²) / 8 = 4.0898

σ = √(4.0898) = 2.022

Now, to estimate the probability that Patsy sells fewer than a certain number of feathers at the next festival, we need to standardize the value by subtracting the mean and dividing by the standard deviation. So, let X be the number of feathers sold at the next festival, then,

Z = (X - μ) / σ

Let's say we want to estimate the probability that Patsy sells fewer than 2 feathers at the next festival. Then,

Z = (2 - 3.375) / 2.022 = -0.679

We can use a standard normal distribution table or calculator to find the probability that a standard normal random variable is less than -0.679, which is approximately 0.2485.

Therefore, a reasonable estimate of the probability that Patsy sells fewer than 2 feathers at the next festival is about 0.2485.

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Correct question is "The number of feathers Patsy 3, 6, 1, 4, 2, 3, 7, 2 Peacock sold at each of the festivals this year. Based on this data, what is a reasonable estimate of the probability that Patsy sells fewer than feathers next festival? "

Answer:

To determine the translation direction and number of units, we need to find the vector that connects Q to Q', and then determine the magnitude and direction of that vector.

The vector that connects Q to Q' can be found by subtracting the coordinates of Q from the coordinates of Q':

Q' - Q = (4, -5) - (4, 3) = (0, -8)

This vector indicates a translation 8 units downwards, in the negative y direction. Therefore, the translation direction is downwards and the number of units is 8.

So the correct answer is: 8 units down.

Triangle PQR was translated 8 units down.

In mathematics, particularly in the field of geometry, a translation refers to moving each point in a shape or a figure to a different position by sliding it to a certain direction for a fixed number of spaces. Each point is moved the same distance and in the same direction.

In the case of your Triangle PQR, the Q point moves from (4, 3) to the new coordinate Q'(4, -5). The x-coordinate in both points remains at 4. Hence, there's no left or right movement. But the y-coordinate changes from 3 to -5. This indicates a downward movement. The distance between 3 and -5 on the number line is 8 units. Therefore, the triangle was translated 8 units down.

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The decimal which is less than 0. 8 and greater than 0. 02 is 0. 46 (option B).

To answer this question, we need to understand how decimals work. A decimal is a number that represents a value less than one, and it is denoted by a dot or a period. The digits that come after the dot represent the number of tenths, hundredths, thousandths, etc., depending on the place value of the digit.

Now, let's consider the given options: 0.81, 0.46, 0.86, and 0.6. We need to find the decimal that lies between 0.8 and 0.02.

We can eliminate option D, 0.6, as it is less than 0.8. Similarly, we can eliminate option A, 0.81, as it is greater than 0.8.

To choose between options B, 0.46, and C, 0.86, we need to compare them with the given values, 0.8 and 0.02. We can see that 0.46 is less than 0.8 but greater than 0.02.

Therefore, the answer is option B, 0.46.

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Answer:

x=25.1

Step-by-step explanation:

by using Pythagorean theorem, [tex]a^{2} +b^{2}=c^{2}[/tex] where c is the hypotenuse

[tex]18.15^{2}[/tex]= 329.4225

[tex]17.39^{2}[/tex]= 302.4121

since those two side lengths are the legs, the sum of the numbers is the length of the hypotenuse squared

therefore, 631.8356 (the sum) = [tex]c^{2}[/tex]

by taking the square root of 631.8356, you will get approx 25.1 for the hypotenuse (aka x)

The slope-intercept equation of line is -1 and equation will be y = -x - 1.

What is the slope?

m = (y2 - y1) / (x2 - x1) is the formula for calculating slope from two points on a line, (x1, y1) and (x2, y2). Here,

m = the line's slope

x1 is equal to the initial point's x-coordinate.

y1 is the first point's y-coordinate.

x2 is equal to the second point's x-coordinate.

The second point's x-coordinate is equal to y2.

x1 = -3, y1 = 4; x2= -2, y2=3

m = (3-4)/(-2 - (-3))

= -1 / (-2+3)

= -1/1

m = -1

Substitute the value in given equation:

y = -x + (-1)

y = -x - 1

Hence, the slope-intercept equation of line is -1 and equation will be y = -x - 1.

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The result of 10 trials expressed in percentage is 70%.

We can utilize the random number table to produce 10 random numbers between 0 and 1 to approximate Nestor's performance in the ten races. If a number is less than or equal to 0.79, it is considered a "medal," and if it is larger than 0.79, it is considered a "no medal." This approach can be repeated ten times to get a sense of the range of possible outcomes.

For ten trials, we get the following results using the random number table shown below:

To estimate the likelihood that Nestor will win at least six of the following ten races, we count how many trials resulted in six or more medals. Seven of the ten trials resulted in six or more medals. As a result, we estimate the likelihood to be 7/10, or 70%.

The likelihood is 70% when expressed as a percentage.

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Standard unit chose to measure the mass of a bowling ball is equal to kilogram .

The unit typically used to measure the mass of a bowling ball is the pound (lb) or the kilogram (kg).

It depends on the country as different countries have different standard unit for measuring mass.

In the United States, the weight of a bowling ball is often measured in pounds.

While in many other countries, the weight is measured in kilograms.

When measuring the mass of a bowling ball,

It is important to use a calibrated scale that is designed to handle the weight of the ball.

Some scales are specifically designed for weighing bowling balls.

And they may have a higher weight capacity than a typical bathroom scale.

It is also important to ensure that the scale is on a flat, stable surface to ensure an accurate measurement.

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The profit function of the company is given by P(x)=-4x^3 + 32x^2 - 64, where x is the number of toys sold in hundreds, and P(x) is the profit in thousands of dollars.

The key features of the graph of the profit function are the following:

The degree of the polynomial function is 3, which means that the graph is a cubic curve.

The coefficient of the leading term is negative (-4), which means that the graph opens downwards.

The coefficient of the quadratic term is positive (32), which means that the graph is concave up.

The y-intercept of the graph is -64, which means that the company will incur a loss of $64,000 if it does not sell any toys.

It should be noted that to find the maximum profit, we need to evaluate the profit function at x = 5.33:

P(5.33) = -4(5.33)^3 + 32(5.33)^2 - 64 = 23.78

Therefore, the maximum profit that the company can make is $23,780.

In summary, the graph of the profit function reveals that the company will incur a loss if it does not sell any toys, but it can make a profit if it sells at least some toys. The profit function has a cubic shape that opens downwards, indicating that the profit decreases as the number of toys sold increases beyond a certain point. The maximum profit occurs at x = 5.33, where the profit is $23,780.

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Answer:

Step-by-step explanation:

Making x the subject in the second equation gives:

x = 4+3y

substituting x = 4 +3y into the first one gives:

3(4 + 3y) + 2y = -21

12 + 9y +2y = -21

11y = -33

y = -3

The distance traveling between them is increasing at a rate of miles per hour = 45 m/h.

The distance between the automobile and the farmhouse is increasing by 45 mph, since the automobile is traveling at this speed.

When the automobile is 9 miles past the intersection of the highway and the road, the distance between the automobile and the farmhouse is increasing at a rate of 45 mph, due to the automobile traveling at this speed.

When the automobile is 9 miles past the intersection of the highway and the road, the distance between the automobile and the farmhouse is increasing at a rate.

So,

The rate at which the distance between the automobile and the farmhouse is increasing when the automobile is 9 miles past the intersection is 45 mph.

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(a) The concentration of the solution in the tank will be changing over time.

(b) The amount of salt in the tank after 3.5 hours is 63.292 kg.

(c) When the inflow and outflow rates are equal, the amount of salt in the tank will remain constant.

(a) To find the concentration of the solution in the tank initially, we can use the formula:

concentration = mass of salt / volume of solution

The mass of salt in the tank initially is 60 kg, and the volume of solution is 2000 liters.

Therefore, the initial concentration is:

concentration = 60 kg / 2000 L

concentration = 0.03 kg/L

However, we know that a solution with a concentration of 0.015 kg/L is entering the tank at a rate of 9 L/min.

Therefore, the concentration of the solution in the tank will be changing over time.

(b) To find the amount of salt in the tank after 3.5 hours, we can use the formula:

amount of salt = initial amount of salt + (concentration of incoming solution - concentration of solution in tank) x rate x time

The initial amount of salt is 60 kg, and the concentration of the incoming solution is 0.015 kg/L.

We need to find the concentration of the solution in the tank after 3.5 hours.

The rate of flow is 9 L/min, so the total volume of solution that has entered the tank after 3.5 hours is:

volume of solution = rate x time

volume of solution = 9 L/min x 210 min

volume of solution = 1890 L

The total volume of solution in the tank after 3.5 hours is:

total volume = initial volume + volume of incoming solution - volume of drained solution

total volume = 2000 L + 9 L/min x 210 min - 9 L/min x 210 min

total volume = 2000 L

Therefore, the concentration of salt in the tank after 3.5 hours is:

amount of salt = 60 kg + (0.015 kg/L - concentration of solution in tank) x 9 L/min x 210 min

amount of salt - 60 kg = (0.015 kg/L - concentration of solution in tank) x 1890 L

concentration of solution in tank = 0.015 kg/L - (amount of salt - 60 kg) / 1890 L

Now we can substitute the concentration of the solution in the tank into the formula and solve for the amount of salt:

amount of salt = 60 kg + (0.015 kg/L - (0.015 kg/L - (amount of salt - 60 kg) / 1890 L)) x 9 L/min x 210 min

amount of salt = 63.292 kg

Therefore, the amount of salt in the tank after 3.5 hours is 63.292 kg.

(c) To find the concentration of salt in the solution in the tank as time approaches infinity, we need to find the concentration that the solution will reach when the inflow and outflow rates of solution are equal.

At this point, the amount of salt in the tank will remain constant.

Let's denote the concentration of salt in the solution in the tank as c.

We know that the volume of solution in the tank remains constant at 2000 L, and that the inflow and outflow rates are both 9 L/min. Therefore, the amount of salt that enters the tank per minute is 0.015 kg/L x 9 L/min = 0.135 kg/min, and the amount of salt that leaves the tank per minute is c x 9 L/min.

When the inflow and outflow rates are equal, the amount of salt in the tank will remain constant.

Therefore, we can set the rate of inflow equal to the rate of outflow and solve for c:

0.015 kg/L x 9.

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The function f(x) is f(x) = 9x³ - 7x² - 13x - 11 written in standard form.

What is the polynomial equation?

A polynomial equation is an equation in which the variable is raised to a power, and the coefficients are constants. A polynomial equation can have one or more terms, and the degree of the polynomial is determined by the highest power of the variable in the equation.

When f(x) is divided by 3x + 1, the quotient is 3x² - 4x - 1 and the remainder is -10. We can use polynomial long division to write f(x) in the form:

f(x) = (3x² - 4x - 1)(3x + 1) - 10

Multiplying out the right side gives:

f(x) = 9x³ - 7x² - 13x - 11

Therefore, the function f(x) is f(x) = 9x³ - 7x² - 13x - 11 written in standard form.

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The exponential function of the form y=ab^x that has the given points is y = (25/7)(7/5)^x

We can use the given points to set up a system of equations and solve for the values of a and b in the exponential function y = ab^x.

Using the point (1,5), we get:

5 = ab^1

Using the point (2,7), we get:

7 = ab^2

Now, we can solve for a and b by dividing the second equation by the first:

7/5 = (ab^2)/(ab^1

Simplifying this expression, we get:

7/5 = b

Substituting this value of b back into one of the original equations, we can solve for a:

5 = a(b^1) = a(b) = a(7/5)

Simplifying this expression, we get:

a = 25/7

So, the exponential function that passes through the points (1,5) and (2,7) is:

y = (25/7)(7/5)^x

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The number of atoms present in 8.500 mole of chlorine atoms can be calculated using Avogadro's number, which is 6.022 x [tex]10^{23}[/tex] atoms per mole. Therefore:

Number of atoms = 8.500 mole x 6.022 x [tex]10^{23}[/tex]atoms/mole

Number of atoms = 5.1177 x [tex]10^{24}[/tex] atoms

The mass of 15.50 mole of oxygen can be calculated using the molar mass of oxygen, which is 16.00 g/mol. Therefore:

Mass = 15.50 mole x 16.00 g/mole

Mass = 248 g

The number of moles of helium in 1.953 x [tex]10^{8}[/tex] g of helium can be calculated using the molar mass of helium, which is 4.00 g/mol. Therefore:

Number of moles = 1.953 x [tex]10^{8}[/tex] g / 4.00 g/mol

Number of moles = 4.883 x [tex]10^{7}[/tex] mol

The number of atoms in 147.82 g of sulfur can be calculated using the molar mass of sulfur, which is 32.06 g/mol, and Avogadro's number. Therefore:

Number of moles = 147.82 g / 32.06 g/mol

Number of moles = 4.608 mol

Number of atoms = 4.608 mol x 6.022 x [tex]10^{23}[/tex] atoms/mol

Number of atoms = 2.773 x [tex]10^{24}[/tex] atoms

The molar mass of Co (cobalt) is 58.93 g/mol.

The formula mass of Ca3(PO4)2 can be calculated by adding the atomic masses of each element in the compound. The atomic masses are:

Ca = 40.08 g/mol

P = 30.97 g/mol

O = 16.00 g/mol

Formula mass = (3 x Ca) + (2 x P) + (8 x O)

Formula mass = (3 x 40.08 g/mol) + (2 x 30.97 g/mol) + (8 x 16.00 g/mol)

Formula mass = 310.18 g/mol

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By Pythagoras Hypotenuse for 1.   9.43 , 2.   9.22 , 3.    9.85 , 4.   12.37, 5.   14.87 , 6.   10.63

7. 9.85 ,   8. 13.04 ,  9. 19.85 ,  10.    7.81

The Pythagorean theorem, commonly referred to as Pythagoras' theorem, is a key relationship in Euclidean geometry between a right triangle's three sides. It declares that the hypotenuse's square, which is the side that is opposite the right angle, is equal to the sum of the squares of the other two sides. In other words, if the hypotenuse is length c and the legs of a right triangle are lengths a and b, then a² + b² = c².

The Pythagorean theorem, which asserts that the square of the hypotenuse is equal to the sum of the squares of the other two sides, can be used to determine the hypotenuse of a right triangle.

This theorem allows us to calculate the hypotenuse value for each of the right triangles presented as follows:

1. a = 5, b = 8

  c = √(a² + b²) = √(5² + 8²) = √(25 + 64) =√(89) ≈ 9.43

2. a = 6, b = 7

  c = √(a² + b²) = √(6² + 7²) = √(36 + 49) = √(85) ≈ 9.22

3. a = 4, b = 9

  c = √(a² + b²) = √(4² + 9²) = √16 + 81) = √(97) ≈ 9.85

4. a = 3, b = 12

  c =√(a² + b²) = √(3² + 12²) = √(9 + 144) = √(153) ≈ 12.37

5. a = 11, b = 10

  c = sqrt(a^2 + b^2) = sqrt(11^2 + 10^2) = sqrt(121 + 100) = sqrt(221) ≈ 14.87

6. a = 8, b = 7

  c = √(a² + b²)= √(8² + 7²) = √(64 + 49) = √(113) ≈ 10.63

7. a = 9, b = 4

  c =√(a² + b²)=√(9² + 4²) = √(81 + 16) = √(97) ≈ 9.85

8. a = 7, b = 11

  c = √(a² + b²)= √(7² + 11²) = √(49 + 121) ≈√(170) ≈ 13.04

9. a=13,b=15

  c=√(a² + b²)=√((13²)+(15²))=√((169)+(225))=√(394))≈19.85

10.a=5,b=6

  c=√(a²+b²)=sqrt((5²)+(6²))=s√((25)+(36))=√(61))≈7.81

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According to unitary method, Beau needs 60 wheels in all to build 9 puppy bots and 6 kitty bots.

The unitary method is a mathematical technique used to solve problems by finding the value of one unit and then calculating the value of the required quantity by multiplying or dividing it with the given value of units.

Given that each bot needs 4 wheels, we can find the number of wheels required to build one bot. Using the unitary method, we can say that one bot needs 4 wheels. Therefore, 9 puppy bots will need 9 times 4 wheels, which is 36 wheels.

Similarly, 6 kitty bots will need 6 times 4 wheels, which is 24 wheels. To find the total number of wheels required to build all the bots, we can add the number of wheels required for puppy bots and kitty bots

Total number of wheels required = 36 + 24 = 60

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Rounding the result off to the nearest 10th, the lateral surface area of the cylinder is 822 inches².

Surface area is the total area of the exposed surfaces of a three-dimensional object. It is measured in square units such as square centimeters (cm2) or square meters (m2). Surface area is an important concept in mathematics, science, and engineering, as it is the total area that determines properties such as friction, heat transfer, and fluid dynamics. For example, a larger surface area can increase the rate of heat transfer and allow for more efficient cooling. Similarly, a larger surface area can increase the friction between two objects, allowing them to grip better. Surface area is also important in chemistry, as it affects the amount of gas or liquid that can be absorbed or released by a given object.

The cylinder has a radius of 11 inches and a height of 12.5 inches. To find the lateral surface area of a cylinder, the formula used is A = 2πrℎ, where r is the radius and h is the height of the cylinder. After plugging in the values, the lateral surface area of the cylinder is 821.75 inches². Rounding the result off to the nearest 10th, the lateral surface area of the cylinder is 822 inches².

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