Sam has a deck that is shaped like a triangle with a base of 18 feet and a height of 7 feet. He plans to build a 2:5 scaled version of the deck next to his horse’s water trough.

Part A: What are the dimensions of the new deck, in feet? Show every step of your work.

Part B: What is the area of the original deck and the new deck, in square feet? Show every step of your work.

Part C: Compare the ratio of the area to the scale factor. Show every step of your work.

(a) The experimental probability of rolling a 3 is approximately 8%.

(b) The experimental probability of rolling a 6 is 25%.

(c) The experimental probability of rolling a number less than 4 is 50%.

The experimental probability of rolling a 3 can be determined from the result presented in the table as shown below.

from the result presented, there a total outcome of 12

number of rolling a 3 in the result = 1

P(3) = 1/12 = 0.083

P(3) ≈ 8%

The experimental probability of a rolling a 6 is calculated as;

number of 6 obtained in the result = 3

P(6) = 3/12

P(6) = 0.25

P(6) = 25%

The experimental probability of rolling a number less than 4:

P ( less than 4) = P(1) + P(2) + P(3)

P ( less than 4) = 17% + 25% + 8%

P ( less than 4) = 50%

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Answer:

A

Step-by-step explanation:

in any right triangle the sine ratio of an angle Θ is

sinΘ = [tex]\frac{opposite}{hypotenuse}[/tex]

One polygon has 6 sides and the other has 7 sides.

Polygons are closed 2D shapes made up of straight lines. Regular polygons have sides of equal length and angles of equal measure. In this problem, we are given information about two regular polygons.

We are told that the total number of sides in both polygons is 13. Let's call the number of sides in one polygon "n" and the number of sides in the other polygon "m". Then, we know that:

n + m = 13

Next, we are given the total number of diagonals in both polygons, which we can calculate using the formula:

d = (n(n-3) + m(m-3))/2

where "d" is the total number of diagonals. We are told that d = 25. Substituting this value and the equation for n + m into the diagonal formula, we get:

25 = (n(n-3) + m(m-3))/2

25 = (n² - 3n + m² - 3m)/2

50 = n² - 3n + m² - 3m

We can use the equation n + m = 13 to solve for one variable in terms of the other. For example, we can solve for "m" by subtracting "n" from both sides:

m = 13 - n

Substituting this into the previous equation, we get:

50 = n² - 3n + (13-n)² - 3(13-n)

50 = n² - 3n + 169 - 26n + n² - 36 + 3n

Simplifying this equation, we get:

2n² - 26n + 45 = 0

We can use the quadratic formula to solve for "n":

n = (26 ± √376)/4

n ≈ 5.61 or n ≈ 8.39

Since we are dealing with whole numbers of sides, "n" must be either 6 or 8. Plugging each value into the equation n + m = 13, we find that the other polygon must have the opposite value.

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The measure οf the center that is mοst apprοpriate tο represent the data in the line plοt graph is -

Optiοn A: The median is the best measure οf center because there are nο οutliers present.

A line plοt is a graph that uses symbοls tο represent data abοve a frequency number line tο indicate the frequency οf each value. It is emplοyed tο arrange the data simply and is very simple tο understand.

Since the data is displayed using a line plοt and the plοt shοws the frequency οf data values, it is clear that the data is discrete.

In this case, the apprοpriate measure οf center wοuld be the median, which is the middle value when the data is arranged in οrder.

The fact that there are sοme dοts abοve certain values suggests that these values οccur mοre frequently in the data set.

Hοwever, this dοes nοt necessarily mean that there are οutliers present. Outliers are extreme values that are far away frοm the majοrity οf the data and can skew the results when using the mean as a measure οf center.

Therefοre, the median is the mοst apprοpriate measure οf center in this case because it is nοt affected by the presence οf οutliers and it represents the middle value οf the data set.

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The value of the identities are;

cos F = 19/20

sin E= √39/20

sin E and cos F are not equal

The different trigonometric identities in mathematics also have their different ratios. They are;

Their ratios are fractions given as;

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

tan θ = opposite/adjacent

From the information given, we have that;

cos F =

Opposite = 19

Hypotenuse = 20

cos F = 19/20

The value of sin E

sin E = 39/20

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If you were not restricted to using just rectangles to fill up the space between the curve and the x-axis, a different geometric shape that might give a more accurate estimate for this area is a trapezoid.

Using trapezoids would likely yield a more accurate result because they can better approximate the curve's shape.
Here's a step-by-step explanation of how using trapezoids can provide a more accurate estimate:

Divide the interval on the x-axis into equal subintervals.

In each subinterval, form a trapezoid by connecting the endpoints of the subinterval to the curve and then closing the shape by drawing a horizontal line from the curve to the x-axis

Calculate the area of each trapezoid using the formula: A = (1/2) * (base1 + base2) * height, where base1 and base2 are the lengths of the parallel sides of the trapezoid (the lengths along the x-axis and the curve), and the height is the length of the vertical line connecting the two bases (the width of the subinterval)

Sum the areas of all trapezoids to approximate the total area between the curve and the x-axis.

The reason trapezoids can provide a more accurate estimate is that they can closely follow the shape of the curve, especially when the curve is not a straight line. Since trapezoids have two different base lengths, they can better accommodate the changes in the curve's slope within each subinterval. This results in a closer fit to the curve compared to using rectangles, which can overestimate or underestimate the area depending on the curve's concavity.

Overall, using trapezoids for estimating the area under a curve can provide more accurate results, especially as the number of subintervals increases.

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5,761.28 square inches of space will require the mirror to take up on the wall when Loretta wants to put a circular mirror on a wall.

We can find the space that the mirror takes up on the wall by using the area of a circle. it is given by:

A = πr^2

where

A = area of a circle

r =radius of a circle

Given data

radius of the mirror = 23 inches

The area of the mirror is:

A = π(23)^2

= 1,661π

where assume π =3.14159:

A = 1,661(3.14159)

A = 5,761.28 square inches

Therefore, the mirror will take up approximately 5,761.28 square inches of space on the wall.

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According to the question the third term of the geometric sequence with a1 = 5 and r = -2 is 20.

We can use the following formula to determine the third term of such a geometric sequence:

a3 = a1 * r^2

where a1 is the first term, r is the common ratio, and a3 is the third term.

In this case, we are given that a1 = 5 and r = -2. With these values entered into in the formula, we obtain:

a3 = 5 * (-2)^2

a3 = 5 * 4

a3 = 20

Since a1 = 5 and r = -2, then third term of a geometric sequence is 20.

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The two square roots of 225 are +15 and -15.

However, since negative values are usually not considered answers for practical purposes, the principal square root of 225 is +15 (i.e., √225 = 15).

Hence, the area of the green sector is roughly [tex]7 cm^2,[/tex] and the arc length of its edge is roughly 4 cm.

Usually expressed in square units, area is a unit of measurement for the size of an a double surface or territory. It is used to indicate the amount of space that lies inside the confines of a straight position, such as a round, rectangle, circle, a triangle. Depending on the shape, a formula is needed to determine the size of a shape. For instance, the area of either a rectangle can be determined by dividing its extent by its width, whereas the area of a circle can be determined by multiplying pi (about 3.14), by the square of its radius. Several branches of mathematics, physics, and engineering all use the idea of area extensively.

given

(Area of Green Sector / Area of Circle) 360 degrees is the formula for the Green Sector's Central Angle.

green sector's centre angle is equal to (7 cm2 / 42 cm2) 360 degrees.

60 degrees is the green sector's midpoint (rounded to the nearest degree)

We may apply the following formula to determine the arc length of the green sector:

circumference (central angle / 360 degrees) = arc length

arc length is equal to 24 cm when divided by 60 degrees.

4-cm-long arc (rounded to one decimal place)

Hence, the area of the green sector is roughly [tex]7 cm^2,[/tex] and the arc length of its edge is roughly 4 cm.

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The work done in the process of lifting coal up a mineshaft that is 632 feet deep is found to be  650,000 ft/lb.

For a cable of density, work = ∫k*y dy from 0 to b.

where k is the linear density of the cable (in lb/ft in this case), y is the distance from the top of the cable, and b is the total distance the weight has to travel to the bottom of the cable.

With this setting, the top of the cable should be labeled y=0 and the bottom of the cable (connected to the coal weight) should be labeled y=b. In this case, k = 2 ft/lb and b = 500 ft.

The 800 lb coal weight is additional information that we have not yet calculated.

Since pounds are a unit of force and the force to lift the coal is constant at 800 pounds, you can add the weight of 800 pounds to the integral and the work done to lift the coal is force * distance . The final integral is:

work = ∫(2y + 800) dy from 0 to 500.

integral:

Work = y2 + 800y | 0 to 500

= (500)2 + 800(500)

So, the work done is 650,000 ft/lb.

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Complete question - A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mine shaft 500 ft deep. Find the work done.

Answer:

The velocity of the upper end P when Q is 4 m from the wall is -0.8 m/s

Step-by-step explanation:

Let's denote the distance of the lower end Q from the wall as x and the distance of the upper end P from the ground as y. We are given that the ladder has a length of 5 m, and thus by the Pythagorean theorem, we have:

x^2 + y^2 = 5^2 = 25

Now, we're given that the lower end Q is sliding away from the wall at a constant rate of 0.6 m/s. This means that the rate of change of x with respect to time (dx/dt) is 0.6 m/s. We need to find the rate of change of y with respect to time (dy/dt) when x = 4 m.

Differentiate both sides of the Pythagorean equation with respect to time (t):

d(x^2)/dt + d(y^2)/dt = d(25)/dt

2x(dx/dt) + 2y(dy/dt) = 0

We can plug in the known values and solve for dy/dt when x = 4 m:

2(4)(0.6) + 2y(dy/dt) = 0

To find the value of y when x = 4 m, we can use the Pythagorean equation:

4^2 + y^2 = 25

16 + y^2 = 25

y^2 = 9

y = 3 (since the height must be positive)

Now, we can substitute y = 3 into the equation we derived earlier:

2(4)(0.6) + 2(3)(dy/dt) = 0

4.8 + 6(dy/dt) = 0

Now, solve for dy/dt:

6(dy/dt) = -4.8

(dy/dt) = -4.8 / 6

(dy/dt) = -0.8 m/s

Thus, the velocity of the upper end P when Q is 4 m from the wall is -0.8 m/s. The negative sign indicates that the upper end P is moving downward.

Answer: Unfortunately, without further context, I can't answer your question. Please post the box plot from the question first.

Answer:

3/5

Step-by-step explanation:

Total= 2x + 2x + x= 5x

Total of Blue and yellow= 2x + x= 3x

So.. 3x/5x which is equivalent to 3/5

F(x,y) · r(x,y) = 4(yi - xj) · (xi + yj)= 4xy - 4xy = 0

The two vectors are perpendicular to each other, and the length of F(x,y) is 4, as desired.

A two-dimensional vector field is a vector field in which each point in the plane is associated with a vector.

It is represented by a pair of functions (P, Q), each of which maps (x, y) to a real number, and the vector field itself is written as F = P i + Q j. Let's write a formula for a two-dimensional vector field that has all vectors of length 4 and is perpendicular to the position vector at that point

The formula for a two-dimensional vector field that has all vectors of length 4 and is perpendicular to the position vector at that point is given by:

F(x,y) = 4(yi - xj)

Here, the position vector at any point (x, y) in the plane is given by r(x,y) = xi + yj, and its magnitude is

|r(x,y)| = √(x² + y²).

To see that F(x,y) is perpendicular to r(x,y), take their dot product:

F(x,y) · r(x,y) = 4(yi - xj) · (xi + yj)= 4xy - 4xy = 0

Thus, the two vectors are perpendicular to each other, and the length of F(x,y) is 4, as desired.

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The total cost of leather bought used to make a football is given as follows:

Rs 1,524.

The surface area of a sphere of radius r is given by the equation presented as follows:

S = 4πr².

FIFA has made a football of a leather of diameter 21 cm, hence the radius is given as follows:

r = 10.5 cm.

Then the surface area is given as follows:

S = 4π x 10.5²

S = 1385.44 cm².

The cost is of Rs 25 per cm², with an additional 10% due to waste, hence the total cost is given as follows:

1.1 x 1385.44 = Rs 1,524.

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The measure of the angle in the angles is 151°

An angle is a figure in Plane Geometry which is formed by two rays or lines that share a common endpoint

the given angles are

<EFH = 44°

and <HFG 107°

This implies that <EFG = <EFH    +  <HFG

= 44 + 107 = 151°

In conclusion, the measure of angle  <EFG = 151°

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A big square with 100 squares in the whole square. 5 of those squares are shaded. The shaded area as both a fraction and a decimal is expressed as:

Fraction: 5/100

Decimal: 0.05

In the form of a numerator and a denominator, fractions express a portion of a whole. Decimals are used to express fractions with a denominator of a power of ten, such as 10, 100, or 1000, in numerical form. Arithmetic is the representation of fractions in their decimal form, where the denominator is 10 or a power of 10 more, such as 100, 1000, or 10,000, etc.

Given:

Area of the whole square = 10 by 10

= 10×10 = 100unit²

The shaded area expressed as a fraction = 5unit²/100unit²

= 51/100

Shaded area expressed as a decimal = 5/100

Then take the number 5 and carry the point by two decimal place from the right

= 0.05

Shaded area expressed as a percent of the whole = (5/100) × 100

= (5×100)/100

= 5%

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The function is both increasing and decreasing on the interval [-2, 1]. The y values decrease from 12 to 0 and then increase to 7.5 through 6. Therefore option B is correct answer.

In mathematics, a function is an expression, rule, or law that establishes the relationship between two variables (the dependent variable). In mathematics, functions are everywhere, and they are crucial for constructing physical relationships in the sciences. German mathematician Peter Dirichlet first offered the modern definition of function in 1837.

Y and x are related in such a way that there is a distinct value of y for each value of x, and this relationship is frequently represented as y = f(x), or "f of x." In other words, the same x cannot have more than one value for f(x). The domain and range of the function are the set of values of f(x) that are produced by the values in the domain of the function, which is the set of values of x. In addition to f(x), other abbreviated symbols for functions of the independent variable x include g(x) and P(x), especially when the nature of the function is ambiguous or unspecified.

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Answer:

6.104

Step-by-step explanation:

Given that,

Mean, {eq}\mu = 6.8 {/eq}

Standard deviation, {eq}\sigma = 0.37 {/eq}

{eq}P(X < x) = 0.03\\ P(\dfrac{X - \mu}{\sigma} < \dfrac{x - \mu}{\sigma}) = 0.03\\ P(Z < \dfrac{x - 6.8}{0.37}) = 0.03\\ P(Z < z) = 0.03 {/eq}

Excel function for the value of z:

=NORMSINV(0.03)

{eq}z = -1.881\\ \dfrac{x - 6.8}{0.37} = -1.881\\ x = 6.8-1.881\times 0.37\\ \color{blue}{x = 6.104 \ minutes} {/eq}

To be sponsored by the running shoe company as one of the fastest 3% of runners, one would need to run faster than the upper 3rd percentile of runners. This can be calculated using the z-score formula which is given as; z = (x - μ) / σ where x is the value we want to find the percentile of, μ is the mean and σ is the standard deviation.

The z-score formula can be rearranged to solve for x which is the running time needed to be sponsored by the company as follows;x = z * σ + μWe want to find the upper 3rd percentile, which corresponds to a z-score of 1.88 (from z-tables or calculator).

Therefore;1.88 = (x - 7.2) / 0.56. Rearranging to solve for x;x = 1.88 * 0.56 + 7.2x = 8.1888 (rounded to 4 decimal places)Therefore, one would need to run faster than 8.1888 minutes to be sponsored by the running shoe company as one of the fastest 3% of runners.

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The graph of the triangle PQR is created by connect the three vertices.

A triangle is a three-sided polygon with three angles. It is a fundamental geometric shape and is often used in geometry and trigonometry. A triangle is a clοsed, 2-dimensiοnal shape with 3 sides, 3 angles, and 3 vertices. A triangle is alsο a pοlygοn.

To graph the triangle PQR, we first plot the three vertices P(-2,4), Q(3,4), and R(-2,-2) on a coordinate plane:

Now, we connect the three vertices to form the triangle PQR:

Thus, the graph of the triangle ΔPQR is shown in the attachment..

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To find the average rate of change of a function over a given interval, we need to calculate the change in the function over that interval, and divide by the length of the interval.

In this case, we want to find the average rate of change of the function f(x) = 12x - 4.8x^2 over the interval [0,5].

First, we need to find the value of the function at the endpoints of the interval:

f(0) = 12(0) - 4.8(0)^2 = 0

f(5) = 12(5) - 4.8(5)^2 = -72

Next, we calculate the change in the function over the interval:

Change in f(x) = f(5) - f(0) = -72 - 0 = -72

Finally, we divide the change in the function by the length of the interval:

Average rate of change = (Change in f(x)) / (Length of interval) = (-72) / (5-0) = -14.4

Therefore, the average rate of change of the function f(x) = 12x - 4.8x^2 over the interval [0,5] is -14.4.

We predict that there will be 512 viruses in the dish after 7 hours.

To model the population growth of the viruses in the dish, we can use a geometric sequence with a common ratio of 2, because the number of viruses doubles every hour.

Let Vn be the number of viruses in the dish after n hours. Then we have:

Recursive formula: Vn = 2Vn-1, with V0 = 4 (since we start with 4 viruses).

Explicit formula: Vn = 4 × 2n.

To predict how many viruses will be in the dish by the 7th hour, we can plug n = 7 into the explicit formula:

V7 = 4 × 27 = 4 × 128 = 512.

Therefore, we predict that there will be 512 viruses in the dish after 7 hours.

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Answer:

Let’s call the base of the triangle b and its height h. The area of a triangle is given by the formula A = 1/2 * b * h.

If Suki says that the area of the triangle is 12 cm², then we can write:

12 = 1/2 * b * h

Similarly, if Emilia says that the area of the triangle is 10 cm², then we can write:

10 = 1/2 * b * h

We can solve for one of the variables in terms of the other by dividing both sides of each equation by 1/2:

b * h = 24

b * h = 20

Now we have two equations with two variables. We can solve for one variable in terms of the other by dividing both sides of one equation by the other:

(b * h) / (b * h) = 24 / 20

1 = 6/5

This is a contradiction, so there is no solution that satisfies both equations.

Step-by-step explanation:

The diameter of the cylinder is approximately 22.98 inches.

To find the diameter of the cylinder, we need to first find the radius using the formula for the volume of a cylinder:
Volume = π × r² × h
Given:
Volume (V) = 2908.33 in³
Height (h) = 7 inches
We will rearrange the formula to find the radius (r):
r² = Volume / (π × h)
Now, plug in the given values:
r² = 2908.33 / (π × 7)
r² ≈ 131.95
Now, find the square root to get the radius:
r ≈ √131.95
r ≈ 11.49 inches
Since the diameter is twice the radius, we can find the diameter (d) by:
d = 2 × r
d = 2 × 11.49
d ≈ 22.98 inches.

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The width of rectangular prism shaped tackle box given by jayce's grandfather to him for use on their first fishing trip together is equals to 8 inches.

We have, jayce's grandfather just give him a new tackle box for using on their first fishing trip together. The shape of box is rectangular prism. Let the dimensions or length, height and width of tackle box be l , h and w. In this case,

Volume of tackle box = 928 in³

Length of tackle box, l = 14.5 inches

Height of tackle box, h = 8 inches

Volume is defined as space occupied by shape. The formula for the volume of a rectangular prism is, Volume (V) = base area × height of the prism or

Volume of rectangular prism, V = l × h × w

So, 928 in³ = l × h × w in³

=> 928 in³ = 14.5 inches × 8 inches × w

=> 928 in³ = 116 in² × w

=> w = 928/116 = 8 inches

Hence, required width is 8 inches.

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Answer:Option(a)

Step-by-step explanation: A) 2 of 15