Two circles, which are tangent externally, are inside and internally tangent to a third circle of radius 1. A diameter of the third circle is a common tangent of the two circles with one point of tangency at its midpoint. What are the radii of the first two circles?

The dean of students at a large college is interested in learning about their opinions regarding the percentage of students who are satisfied with their academic experience. This is an important question for the dean to consider, as it can help them identify areas for improvement and ensure that they are meeting the needs of their students.

To gather this information, the dean may choose to conduct a survey or hold focus groups with students to hear their feedback. They may also review data on student retention rates and academic performance to get a sense of how satisfied students are with their experience at the college.

Once the dean has gathered this information, they can use it to make informed decisions about how to improve the academic experience for their students. This may involve investing in new programs or resources to better support students, or making changes to existing programs based on feedback from students.

Ultimately, by taking the time to gather and listen to student opinions about their academic experience, the dean can create a more student-centered learning environment that meets the needs and expectations of their diverse student population.

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The lateral area of the cone is 18918 m²

The lateral area of the cone can be determined using the formula:

A[tex]_{L}[/tex] = πrL

Where is the r is the radius of circular base of the cone and L is the slant height

In this case:

r = 140/2 = 70m

L = √(50² + 70²)   (Pythagoras theorem)

L = 10√74 m

A[tex]_{L}[/tex] = π * 70 * 10√74

A[tex]_{L}[/tex] = 18918 m²

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Answer:For 19 to 250 odds against winning;

Probability of:

Winning = (0.9294) or 92.9368%

Losing = (0.0706) or 7.0632%

"Odds for" winning: 250:19

"Odds against" winning: 19:250

Step-by-step explanation:

The area of glass value is 0.18 square meters.

The longer diagonal of the kite-shaped window has a measure of 25 cm + 35 cm = 60 cm.

The shorter diagonal has a measure of 30 cm + 30 cm = 60 cm as well.

To find the area of the window, we can use the formula -

Area = (1/2) × d1 × d2

where d1 and d2 are the lengths of the longer and shorter diagonals, respectively.

Substituting the values, we get -

Area = (1/2) × 60 cm × 60 cm

Area = 1800 square cm

However, the question asks for the area in square meters, so we need to convert square centimeters to square meters by dividing by 10,000 -

Area = 1800 / 10,000 square meters

Area = 0.18 square meters

Therefore, the area of glass value is 0.18 square meters.

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The answer is option C) 30,700

The problem involves using linear interpolation to estimate the population of Little Elm in 2020 based on the given data from 2010 and 2015.

To start, we need to calculate the annual growth rate of the population between 2010 and 2015. This can be done by taking the difference between the population in 2015 and the population in 2010, and then dividing that difference by the number of years in that time period:

Annual growth rate = (17,150 - 3,646) / 5 = 2,700

This means that, on average, the population of Little Elm grew by 2,700 people per year between 2010 and 2015.

Next, we can use this annual growth rate to estimate the population of Little Elm in 2020. To do this, we need to add 10 years of growth to the initial population of 3,646 (since we want to estimate the population in 2020, which is 10 years after 2010):

Estimated population in 2020 = 3,646 + (2,700 x 10) = 30,646

Therefore, the estimated population of Little Elm in 2020 is approximately 30,700, which corresponds to answer choice C.

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The estimate from direct proportionality is higher than the prediction from KLEIBER's law.

a. For a 180 kilogram lion, we can use KLEIBER's law: R = 73.3*(180^0.75) = 6136.5 Calories per day. For an 80 kilogram human, we can use the same formula: R = 73.3*(80^0.75) = 1537.6 Calories per day.

b. If metabolic rate and mass were directly proportional, we could use the ratio of the masses to find the basal metabolic rate for an 80 kilogram human.

The ratio of the masses is 80/180 = 0.44. We can multiply this by the basal metabolic rate for the lion to get an estimate for the human: 0.44*6136.5 = 2701.26 Calories per day.

This estimate is higher than the prediction from Kleiber's law for an 80 kilogram human (1537.6 Calories per day). KLEIBER's law indicates that smaller organisms have a higher metabolic rate per kilogram of mass than do larger organisms.

Therefore, the estimate from direct proportionality is higher than the prediction from KLEIBER's law.

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True. In a continuous uniform distribution, all values within a specified interval are equally likely to occur. The mean or expected value of this distribution can be found by taking the average of the upper and lower limits of the interval.

This can be represented mathematically as (a+b)/2, where a and b are the lower and upper limits of the interval. For example, if we have a continuous uniform distribution on the interval [0,10], the mean value would be (0+10)/2 = 5.

This means that on average, we would expect a randomly selected value from this distribution to be around 5. It's important to note that the mean of a continuous uniform distribution is not affected by the shape or spread of the distribution, as all values have an equal chance of occurring.

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The situation provided is exponential growth and the equation is

p = 225 * 1.133ˣ

This scenario represents a significant improvement as the number of students increases each year.

The basic approach to incremental growth is:

[tex]P = P_{0} * (1 + r)^t[/tex]

where:

P. = initial population

r = increase in decimal form

t = time in years

Here

P₀ = 225 (initial population in 2021).

r = 13.3% = 0.133 (growth rate as decimal) .

t = x (time in years from the opening of the school in 2021)

Substituting these values ​​in the formula we get:

[tex]p = 225 * (1 + 0.133)^x[/tex]

Simplifying further, we get:

p = 225 * 1.133ˣ

This is the equation that predicts school population x years after the school opens in 2021

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The estimated volume of the snowpack in the park is approximately 1.94 billion cubic meters.

To estimate the volume of the snowpack in the park, we need to calculate a double integral of the function d(x, y) over the domain D:

V = ∬_D d(x, y) dA

We can evaluate this integral by using iterated integrals. First, we integrate with respect to x over the interval [0, 30] (since the line y = 180 - 2x intersects the x-axis at x = 90):

V = ∫_0^30 (∫_0^(180-2x) (-0.024x + 0.012y + 1.2) dy) dx

Simplifying the inner integral:

V = ∫_0^30 [-0.024xy + 0.006y^2 + 1.2y]_0^(180-2x) dx

V = ∫_0^30 (-0.432x^2 + 21.6x - 5832) dx

Simplifying the integral:

V = [-0.144x^3 + 10.8x^2 - 5832x]_0^30

V = -0.144(30)^3 + 10.8(30)^2 - 5832(30)

V = 1.942592 billion cubic meters (rounded to nine decimal places)

Therefore, the estimated volume of the snowpack in the park is approximately 1.94 billion cubic meters.

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Mato will pay about $700 more in interest than Kika ($625 - $1,300 = $675, which rounds to $700). The answer is C: Mato; $700

Mato will pay more in interest because he has a higher interest rate and a shorter repayment period. To calculate the amount of interest each will pay, we can use the formula:

Interest = (Loan amount) x (Interest rate) x (Time in years)

For Kika:
Interest = $5,000 x 0.052 x 5
Interest = $1,300

For Mato:
Interest = $5,000 x 0.075 x (2/12)
Interest = $625

Therefore, Mato will pay about $700 more in interest than Kika ($625 - $1,300 = $675, which rounds to $700). The answer is C: Mato; $700.

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The Equation of the line which is parallel to the line "y = 3x + 6", and also passes through (4,7) is "y = 3x - 5".

In order to find an equation of a line which passes through point (4,7) and is parallel to line "y = 3x + 6", we use the fact that parallel lines have the same slope.

The given line ""y = 3x + 6" has a slope of 3,

So, the "parallel-line" we want to find must also have a slope of 3.

Now, by using the "point-slope" form of the equation of a line, which is : y - y₁ = m(x - x₁),

where m is slope and (x₁, y₁) is a point on the line,

So, we substitute "m = 3" and (x₁, y₁) = (4,7) to get:

⇒ y - 7 = 3(x - 4),

⇒ y - 7 = 3x - 12,

⇒ y = 3x - 5

Therefore, the equation of the required line is y = 3x - 5.

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The Colorado hikers are at an altitude of 8753.1694 feet.

Here we have been given the equation T = -1. 83a + 212.

Here, T is the boiling point in Fahrenheit of water, while a is the altitude of the place.

We know that the campers in hiing in Colorado boil water for ea. The water boils at a temperature of 196°F, we need to find the altitude.

Here we are going to clearly take T = 196°F

Substituting the value in the above equation will give us

196 = - 1.83a + 212

Taking the variable to the Right Hand Side will give us

196 + 1.83a = 212

Taking 196 to LHS to separate the constant and variable will give us

1.83a = 212 - 196

Solving LHS gives us

1.83a = 16

Taking 1.83 to the other side will give us

a = 16/1.83

or, a = 8.7431694 thousands feet

= 8753.1694 feet

Hence, the Colorado hikers are at an altitude of 8753.1694 feet.

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Answer:

Here is the complete magic square:

-3 4 -1

2 0 -2

1 -4 3

Every row, column, and long diagonal adds to 0.

The volume of the prism is 78 cubic feet.The volume of a prism is given by the formula V = Bh, where B is the area of the base and h is the height of the prism.

In this case, the base of the prism is a right triangle with legs measuring 3 feet and 4 feet, so its area is (1/2)(3)(4) = 6 square feet. The height of the prism is 13 feet. Therefore, the volume of the prism is V = Bh = (6)(13) = 78 cubic feet.To understand this calculation, think of the prism as a stack of identical, parallel cross sections. Each cross section is a copy of the base, with an area of 6 square feet.

The height of each cross section is the same, and equal to the height of the prism, which is 13 feet. To find the total volume of the prism, we add up the volumes of all these cross sections, which is equal to the area of the base times the height of the prism.

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Answer:

--------------------

Let the original number be n.

We are given that 30% of n is 14.99.

Set up an equation to reflect this relation:

Solve it for n:

The standard form of the quadratic relation y=2(x-2)^2-18 is y=2x^2-8x-14.

The standard form of a quadratic relation is y=ax^2+bx+c, where a, b, and c are constants. To convert y=2(x-2)^2-18 to standard form, we need to expand the squared term and simplify the expression.

First, we expand the squared term to get y=2(x^2-4x+4)-18. Then, we distribute the 2 to get y=2x^2-8x+8-18. Finally, we simplify by combining like terms to get the standard form y=2x^2-8x-14.

This quadratic relation is a parabola with a vertex at (2,-18) and it opens upwards since the coefficient of x^2 is positive. The axis of symmetry is a vertical line passing through the vertex, and the y-intercept is -14.

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A circumscribed angle is an angle whose sides are tangent to a circle. In other words, the angle is formed by two intersecting chords of the circle. The vertex of the angle is located outside of the circle, while the two endpoints of the angle lie on the circle.

Circumscribed angles have some important properties in geometry. For example, the measure of a circumscribed angle is half the measure of its intercepted arc (the arc of the circle that lies inside the angle). Additionally, if two angles intercept the same arc of a circle, they are congruent.

Circumscribed angles also appear frequently in trigonometry, where they are used to define the sine, cosine, and tangent functions. The sine of a circumscribed angle is defined as the ratio of the length of the opposite side of the angle to the length of the circle's radius. The cosine of a circumscribed angle is defined as the ratio of the length of the adjacent side of the angle to the length of the radius.

Finally, the tangent of a circumscribed angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

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The required inequalities are h + l ≤ 20, 10h + 15l ≥ 240 and 20h + 25l ≥ 440

Given, for tutoring elementary students in math Susan earns $10 per hour. She earns $15 per hour for cleaning in the library.

Let h be the number of hours Susan works in one week tutoring elementary students.

Let l be the number of hours Susan works in one week cleaning the library.

Given that each week Susan cannot work more than 20 hours.

So, h + l ≤ 20 ....(1)

Susan's total earnings must be at least $240 per week.

10h + 15l ≥ 240    ...(2)

Multiplying equation (1) by 10

10h + 10l ≤ 200    ...(3)

Adding equations (2) and (3)

20h + 25l ≥ 440

Thus, the three required inequalities are h + l ≤ 20, 10h + 15l ≥ 240 and 20h + 25l ≥ 440

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The time taken by the younger sister to finish the same work by herself is 12 hours.

To solve the problem of how long it will take the younger sister to finish the job by herself, let's use the following terms:

1. Older sister's work rate = O
2. Younger sister's work rate = Y
3. Time taken by the younger sister alone = T

Given that the older sister works 3 times faster than the younger sister, we have:  O = 3Y.

Also, the sisters together can finish the job in 3 hours. Therefore, their combined work rate is equal to completing 1/3 of the job per hour. So,

O+Y=1/3.

Now, we can substitute O with 3Y:  3Y+Y=1/3. Combine the terms and simplify:

4Y=1/3

Now, solve for Y:

Y=1/12

Since Y is the work rate of the younger sister, to find the time it takes for her to complete the job alone (T), we can use the following formula:

Work rate × Time = 1 job.

So, Y × T = 1.

Substitute Y with  1/12:

[tex]\frac{1}{12} \times T=1[/tex]

Now, solve for T:

T = 12.

Therefore, it will take the younger sister 12 hours to finish the job by herself.

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Based on the inscribed angle theorem, the measure of angle JKL in the circle is: m<JKL = 121°.

Where an inscribed angle is subtended by an arc it intercepts, the measure of the inscribed angle is equal to half of the the measure of the intercepted arc in the circle, based on the inscribed angle theorem.

Angle JKL is the inscribed angle that is subtended by arc JML. Find the measure of arc JML.

Measure of arc JML = 360 - 53 - 65

Measure of arc JML = 242°

m<JKL = 1/2(measure of arc JML) [inscribed angle theorem]

Substitute:

m<JKL = 1/2(242)

m<JKL = 121°

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Write the quadratic equation in standard form that represents the volume of the box:[tex]w^2 - 6w = 0[/tex]

Solve the quadratic equation for x and find the length and width of the box:

w(w - 6) = 0 (factor the quadratic)

w = 0 or w = 6 (apply the zero product property)

Since a box can't have a width of 0, we reject the solution w = 0.

So, the only solution is w = 6.

To find the length, we use the formula l = V/wh:

l = 6/(6h) = 1/h

The length depends on the value of h, but we can choose h = 1/6 ft to get a reasonable set of dimensions:

length = 1 ft

width = 6 ft

height = 1/6 ft

Therefore, the quadratic equation that represents the volume of the box is[tex]w^2 - 6w = 0[/tex], and the dimensions of the box are length = 1 ft, width = 6 ft, and height = 1/6 ft.

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When we convert the given 54 square feet into square yards we get 6 square yards by using two unit multipliers to convert.

A fraction that equals 1 and is used to convert one set of units to another one is a unit multiplier. The fraction numerator and denominator contain equivalent measurements in different units.

We need to convert the 54 square feet to square yards by using two unit multipliers. by using the two-unit multipliers

given standards :

1 yard = 3 feet

1 square yard = 9 square feet

To convert the 54 square feet to square yards we need to multiply 54 square feet by two unit multipliers which are (1 yard / 3 feet) and (1 yard / 3 feet). Then the equation can be written as:

= 54 square feet × (1 yard / 3 feet) × (1 yard / 3 feet)

= 6 square yards

Therefore, 54 square feet is equal to 6 square yards.

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The ladder must be situated at an angle of approximately 63.43° to reach the top of the 9 ft wall.

How to find the angle of elevation at which the ladder must be situated?

Certainly, here's an illustration of the problem:

           |\

           | \

           |   \  9 ft

           |     \

ladder |       \

  (10 ft)|_____\

      wall

To find the angle of elevation at which the ladder must be situated, we can use the trigonometric function of sine. Let θ be the angle of elevation. Then:

sin θ = opposite / hypotenuse

In this case, the opposite side is the height of the wall (9 ft), and the hypotenuse is the length of the ladder (10 ft). So:

sin θ = 9/10

Using a calculator or a trigonometric table, we can find the angle whose sine is 9/10:

θ ≈ 63.43°

Therefore, the ladder must be situated at an angle of approximately 63.43° to reach the top of the 9 ft wall.

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The probability of drawing a diamond or a spade card from a standard deck of cards and rolling a 2 on a six-sided die is 8.33%

To calculate the probability of drawing a diamond or a spade card from a standard deck of cards, we need to find the total number of diamond and spade cards in the deck. There are 13 cards in each suit, so there are 26 diamond and spade cards in total. The deck has 52 cards in total, so the probability of drawing a diamond or a spade card is:

P(diamond or spade) = 26/52 = 1/2 = 50%

To calculate the probability of rolling a 2 on a six-sided die, we need to find the total number of possible outcomes, which is 6 (since there are 6 sides on the die), and the number of favorable outcomes, which is 1 (since there is only one face with a 2 on it). Therefore, the probability of rolling a 2 on a six-sided die is:

P(rolling a 2) = 1/6 = 16.67%

To find the probability of both events happening together (drawing a diamond or a spade card and rolling a 2 on a six-sided die), we multiply the probabilities of each event:

P(diamond or spade AND rolling a 2) = P(diamond or spade) * P(rolling a 2)
= 50% * 16.67%
= 8.33%

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The answer is 2143.6.

The x-coordinates of all inflection points a f of the function[tex]f(x) = -x^4 - 8x^3 + 6x - 2[/tex]are (-4, f(-4)) and (0, f(0)).

To find the intervals of concavity and the inflection points of the function[tex]f(x) = -x^4 - 8x^3 + 6x - 2,[/tex] we need to find the second derivative and analyze its sign.

First, we find the first derivative:

[tex]f'(x) = -4x^3 - 24x^2 + 6[/tex]

Then, we find the second derivative:

[tex]f''(x) = -12x^2 - 48x[/tex]

To determine the intervals of concavity, we need to find where f''(x) is positive or negative.

[tex]f''(x) = -12x^2 - 48x = -12x(x + 4)[/tex]

f''(x) is negative for x < -4 and x > 0, and positive for -4 < x < 0.

Therefore, the function f(x) is concave down on the intervals (-∞, -4) and (0, ∞), and concave up on the interval (-4, 0).

To find the inflection points, we need to find where the concavity changes. This occurs at x = -4 and x = 0.

At x = -4, the function changes from concave down to concave up. Therefore, (-4, f(-4)) is an inflection point.

At x = 0, the function changes from concave up to concave down. Therefore, (0, f(0)) is also an inflection point.

Thus, the inflection points of f(x) are (-4, f(-4)) and (0, f(0)).

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The volume of the material used to make the insulated collar is 1060.56 cubic inches.

The volume of the material used to make the insulated collar can be determined as follows.

1. Identify the given dimensions:

inner radius (r) = 3 inches,

outer radius (R) = 5 inches, and

height (h) = 21 inches.

2. Use the formula for the volume of a cylinder:

V = πr^2h.

3. Calculate the volume of the outer cylinder with radius R:

V_outer = πR^2h = 3.14 * (5^2) * 21 ≈ 1653.3 cubic inches.

4. Calculate the volume of the inner cylinder with radius r:

V_inner = πr^2h = 3.14 * (3^2) * 21 ≈ 592.74 cubic inches.

5. Subtract the volume of the inner cylinder from the volume of the outer cylinder to find the volume of the material used:

V_material = V_outer - V_inner ≈ 1653.3 - 592.74 ≈ 1060.56 cubic inches.

The volume of the material is approximately 1060.56 cubic inches, rounded to the nearest hundredth.

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The length of the missing side is equal to 18 inches.

In Mathematics and Geometry, the perimeter of a triangle can be calculated by using this mathematical equation:

P = a + b + c

Where:

By substituting the given parameters or dimensions into the formula for the perimeter of a triangle, we have the following;

51 = 18 + 15 + x

51 = 33 + x

x = 51 - 33

x = 18 inches.

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