Roger can run one mile in 9 minutes. Jeff can run one mile in 6 minutes. If Jeff gives Roger a 1 minute head start, how
long will it take before Jeff catches up to Roger? How far will each have run?
They each will have run of a mile.

The volume of ice cream that can fit within the cone is 84.78 cubic inches.

To determine the volume of ice cream that can fit within the cone, we can consider the cone as a right circular cone with a height of 9 inches and a radius of 3 inches (half of the diameter, which is 6 inches).

The formula for the volume of a right circular cone is given by:

V = (1/3) * π * r^2 * h

where V represents the volume, π is a mathematical constant (approximately 3.14159), r is the radius, and h is the height.

Substituting the given values into the formula, we have:

V = (1/3) * 3.14159 * 3^2 * 9

= (1/3) * 3.14159 * 9 * 9

≈ 84.78 cubic inches

Therefore, the volume of ice cream that can fit within the cone is approximately 84.78 cubic inches.

To provide a visual representation, imagine a cone shape with a height of 9 inches and a diameter of 6 inches. The radius is half of the diameter, so it is 3 inches.

The ice cream fills the cone up to the top, creating a rounded triangular shape. The volume of the ice cream is equivalent to the volume of this cone shape.

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Note the complete question is

Bela's ice cream cone is 9 inches tall and 6 inches across. What volume of ice cream can fit within the cone? Show your work and draw a picture of the scenario.

Type your answer as a number only.

Round your answer to the nearest tenth. Volume = cubic inches

The result of the row operation on the matrix is given as follows:

[tex]\left[\begin{array}{cccc}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

The matrix in this problem is defined as follows:

[tex]\left[\begin{array}{cccc}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

The row operation is given as follows:

[tex]R_1 \rightarrow \frac{1}{2}R_1[/tex]

The first row of the matrix is given as follows:

[2 0 0 16]

The meaning of the operation is that every element of the first row of the matrix is divided by two.

Hence the resulting matrix is given as follows:

[tex]\left[\begin{array}{cccc}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

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Triangles ZWN and ZXY are similar by the SAS congruence theorem.

The Side-Angle-Side (SAS) congruence theorem states that if two sides of two similar triangles form a proportional relationship, and the angle measure between these two triangles is the same, then the two triangles are congruent.

In this problem, we have that the angle Z is equals for both triangles, and the two sides between the angle Z, which are ZW = ZY and ZV = ZX, form a proportional relationship.

Hence the SAS theorem holds true for the triangle in this problem.

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Answer: False

Step-by-step explanation:

Spherical geometry, on the other hand, is a type of non-Euclidean geometry that is specifically concerned with studying the properties of curved surfaces, such as spheres.

Hope this help! Have a good day!

Hello!

x² - 5x + 6

= (x² - 2x) + (-3x + 6)

= x(x - 2) - 3(x - 2)

= (x - 2)(x - 3)

The correct Option is D. 45÷3= 15 . The equation that best shows that 45 is a multiple of 15 is 45 ÷ 3 = 15.

A multiple is a product that results from multiplying two or more numbers.

A common multiple is a multiple that is common to two or more numbers.

A multiple of a number can be expressed as an integer multiple of the number.

If the result is a whole number, the first number is a multiple of the second.

An equation that shows 45 is a multiple of 15 is as follows: 45 ÷ 3 = 15.

A multiple is a number that can be divided by another number without leaving a remainder.

As a result, we divide 45 by 3 to find out whether 45 is a multiple of 15.

If the result is a whole number, 45 is a multiple of 15.

Here is the equation that shows this: 45 ÷ 3 = 15

Thus, we can conclude that the equation that best shows that 45 is a multiple of 15 is  45 ÷ 3 = 15.

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Answer:

0

Step-by-step explanation:

[tex]\displaystyle \frac{F(b)-F(a)}{b-a}\\\\=\frac{F(\pi)-F(-\pi)}{\pi-(-\pi)}\\\\=\frac{\sin(4\pi)-\sin(-4\pi)}{2\pi}\\\\=\frac{0-0}{2\pi}\\\\=0[/tex]

Not sure how the correct answer is stated as 45/28, but the answer is definitely 0.

The values in the domain of fᵒg are all real numbers.

Therefore, the correct answer is: x - 6.

To determine the domain of the composite function fᵒg, we need to find the values of x that are valid inputs for the composition.

The composite function fᵒg represents applying the function f to the output of the function g. In this case, g(x) is equal to -9.

So, we substitute -9 into the function f(x) = √6x:

f(g(x)) = f(-9) = √6(-9) = √(-54)

Since the square root of a negative number is not defined in the set of real numbers, the value √(-54) is undefined.

Therefore, -9 is not in the domain of fᵒg.

To find the values in the domain of fᵒg, we need to consider the values of x that make g(x) a valid input for f(x).

Since g(x) is a constant function equal to -9, it does not impose any restrictions on the domain of f(x).

The function f(x) = √6x is defined for all real numbers, as long as the expression inside the square root is non-negative.

So, any value of x would be in the domain of fᵒg.

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Answer:

1

Step-by-step explanation:

The degree of a polynomial is the exponent of the varieble x. So the exponent of x in that function is 1.

The rule that makes the machine work is *-5 + 6 * -5

From the question, we have the following parameters that can be used in our computation:

4      -50

-8      10

-3     -15

A linear equation is represented as

y = mx + c

Using the points, we have

4m + c = -50

-8m + c = 10

Subtract the equations

So, we have

12m = -60

m = -5

Next, we have

-8 * -5 + c = 10

So, we have

c = 10 - 40

c = -30

This means that the operation is

-5x - 30

When expanded, we have

*-5 + 6 * -5

Hence, the rule that makes the machine work is *-5 + 6 * -5

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The area of the obtuse triangle is 34 square feet with a base of 10 ft and height of 6.8 ft.

To find the area of the obtuse triangle, we can use the formula A = (1/2) * base * height. Let's denote the unknown part of the base as x.

In the given triangle, we have the width of the obtuse angle triangle as 10 ft, the height (perpendicular) as 6.8 ft, and the unknown part of the base as x.

Using the formula, we can calculate the area as:

A = (1/2) * (10 + x) * 6.8

Simplifying this expression, we get:

A = 3.4(10 + x)

Now, we need to determine the value of x. From the given information, we know that the width of the obtuse angle triangle is 10 ft. This means the sum of the two parts of the base is 10 ft. Therefore, we can write the equation:

x + 10 = 10

Solving for x, we find:

x = 0

Since x = 0, it means that one part of the base has a length of 0 ft. Therefore, the entire base is formed by the width of the obtuse angle triangle, which is 10 ft.

Now, substituting this value of x back into the area formula, we have:

A = 3.4(10 + 0)

A = 3.4 * 10

A = 34 square feet

Hence, the area of the obtuse triangle is 34 square feet.

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The tangent and normal lines of the curve:

Case (i): y = (4 / 5) · x + 13 / 5

Case (ii): (x, y) = (0, 13 / 5)

Case (iii): y = - (5 / 4) · x + 35 / 4

In this problem we have the representation of a curve whose equations for tangent and normal lines must be found. Lines are expressions of the form:

y = m · x + b

Where:

Both tangent and normal lines are perpendicular, the relationship between the slopes of the two perpendicular lines is:

m · m' = - 1

Where:

The slope of the tangent line is found by evaluating the first derivative of the curve at intersection point.

Case (i) - First, determine the slope of the tangent line:

y = √(8 · x + 1)

y' = 4 / √(8 · x + 1)

y' = 4 / √25

y' = 4 / 5

Second, determine the intercept of the tangent line:

b = y - m · x

b = 5 - (4 / 5) · 3

b = 5 - 12 / 5

b = 13 / 5

Third, write the equation of the tangent line:

y = (4 / 5) · x + 13 / 5

Case (ii) - Find the coordinates of the intercept of the tangent line:

(x, y) = (0, 13 / 5)

Case (iii) - First, find the slope of the normal line:

m' = - 1 / (4 / 5)

m' = - 5 / 4

Second, determine the intercept of the normal line:

b = y - m' · x

b = 5 - (- 5 / 4) · 3

b = 5 + 15 / 4

b = 35 / 4

Third, write the equation of the normal line:

y = - (5 / 4) · x + 35 / 4

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The length of line segment IG of the circle using the chord-chord power theorem is 6.

Chord-chord power theorem simply state that "If two chords of a circle intersect, then the product of the measures of the parts of one chord is equal or the same as the product of the measures of the parts of the other chord".

From the figure:

Line segment FG = 12

Line segment GH = 4

Line segment GJ = 8

Line segment IG = ?

Now, usig the chord-chord power theorem:

Line segment FG × Line segment GH = Line segment GJ × Line segment IG

Plug in the values:

12 × 4 = 8 × Line segment IG

48 = 8 × Line segment IG

Line segment IG = 48/8

Line segment IG = 6

Therefore, the line segment IG measures 6 units.

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Answer:

y^10 + (5/8xy^5 - 5/8xy^6) - (25/64x^2)

Step-by-step explanation:

To simplify the given expression, we can expand it using the distributive property:

(5/8x + y^5)(y^5 - 5/8x)

Expanding the expression yields:

= (5/8x * y^5) + (5/8x * -5/8x) + (y^5 * y^5) + (y^5 * -5/8x)

= (5/8xy^5) - (25/64x^2) + y^10 - (5/8xy^6)

Combining like terms, we have:

= y^10 + (5/8xy^5 - 5/8xy^6) - (25/64x^2)

Hope this help! Have a good day!

5 whole numbers are written in order. 5,8,x,y,12 The mean and median of the five numbers are the same then the values of x and y are:[tex]$$\boxed{x=8, \ y=3}$$[/tex] OR [tex]$$\boxed{x=12, \ y=53}$$[/tex].

let's first calculate the median of the given numbers.

Median of the given numbers is the middle number of the ordered set.

As there are five numbers in the ordered set, the median will be the third number.

Thus, the median of the numbers = x.

The mean of a set of numbers is the sum of all the numbers in the set divided by the total number of items in the set.

Let the mean of the given set be 'm'.

Then,[tex]$$m = \frac{5+8+x+y+12}{5}$$$$\Rightarrow 5m = 5+8+x+y+12$$$$\Rightarrow 5m = x+y+35$$[/tex]

As per the given statement, the median of the given set is the same as the mean.

Therefore, we have,[tex]$$m = \text{median} = x$$[/tex]

Substituting this value of 'm' in the above equation, we get:[tex]$$x= \frac{x+y+35}{5}$$$$\Rightarrow 5x = x+y+35$$$$\Rightarrow 4x = y+35$$[/tex]

Also, as x is the median of the given numbers, it lies in between 8 and y.

Thus, we have:[tex]$$8 \leq x \leq y$$[/tex]

Substituting x = y - 4x in the above inequality, we get:[tex]$$8 \leq y - 4x \leq y$$[/tex]

Simplifying the above inequality, we get:[tex]$$4x \geq y - 8$$ $$(5/4) y \geq x+35$$[/tex]

As x and y are both whole numbers, the minimum value that y can take is 9.

Substituting this value in the above inequality, we get:[tex]$$11.25 \geq x + 35$$[/tex]

This is not possible.

Therefore, the minimum value that y can take is 10.

Substituting y = 10 in the above inequality, we get:[tex]$$12.5 \geq x+35$$[/tex]

Thus, x can take a value of 22 or less.

As x is the median of the given numbers, it is a whole number.

Therefore, the maximum value of x can be 12.

Thus, the possible values of x are:[tex]$$\boxed{x = 8} \text{ or } \boxed{x = 12}$$[/tex]

Now, we can use the equation 4x = y + 35 to find the value of y.

Putting x = 8, we get:

[tex]$$y = 4x-35$$$$\Rightarrow y = 4 \times 8 - 35$$$$\Rightarrow y = 3$$[/tex]

Therefore, the values of x and y are:[tex]$$\boxed{x=8, \ y=3}$$[/tex]  OR [tex]$$\boxed{x=12, \ y=53}$$[/tex]

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Answer:

Step-by-step explanation:

its on satuday

Answer:

Step-by-step explanation:

The approximate age of the rock outcrop is 1.2 billion years.

To approximate the age of the rock outcrop, we can use the concept of radioactive decay and the half-life of the U-238 isotope.

The half-life of U-238 is 4.5 billion years, which means that after each half-life, the amount of U-238 remaining is reduced by half.

We are given that the rock outcrop has 89.00% of its parent U-238 isotope remaining.

This means that the remaining fraction is 0.8900.

To find the number of half-lives that have elapsed, we can use the following formula:

Number of half-lives = log(base 0.5) (fraction remaining)

Using this formula, we can calculate:

Number of half-lives = log(base 0.5) (0.8900)

≈ 0.1212

Since each half-life is 4.5 billion years, we can find the approximate age of the rock outcrop by multiplying the number of half-lives by the half-life duration:

Age of the rock outcrop = Number of half-lives [tex]\times[/tex] Half-life duration

≈ 0.1212 [tex]\times[/tex] 4.5 billion years

≈ 545 million years

Therefore, the approximate age of the rock outcrop is approximately 545 million years.

Based on the answer choices provided, the closest option to the calculated value of 545 million years is 757 million years.

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The function representing the situation is f(x) = 2(x + 1)^2 + 2. Option C.

To determine which function represents the given situation, we can use the information provided about the minimum point and the point on the parabola.

We are given that the minimum of the graph of the quadratic function is located at (-1, 2). This means that the vertex of the parabola is at (-1, 2).

The standard form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.

In this case, h = -1 and k = 2, so the equation becomes f(x) = a(x + 1)^2 + 2.

Additionally, we know that the point (2, 20) lies on the parabola. We can substitute these coordinates into the equation to solve for the value of a:

20 = a(2 + 1)^2 + 2

20 = 9a + 2

18 = 9a

a = 2

Substituting the value of a back into the equation, we have:

f(x) = 2(x + 1)^2 + 2

So the function that represents the given situation is f(x) = 2(x + 1)^2 + 2, which is option C.

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Note the complete question is

The minimum of the graph of a quadratic function is located at (–1, 2). The point (2, 20) is also on the parabola. Which function represents the situation?

A) f(x) = (x + 1)2 + 2

B) f(x) = (x – 1)2 + 2

C) f(x) = 2(x + 1)2 + 2

D) f(x) = 2(x – 1)2 + 2

Answer:

The answer is, x= 145

Step-by-step explanation:

Since line BD passes through the center E of the circle, then the angle must be a right angle or a 90 degree angle.

Hence angle DAB must be 90 degrees

or,

[tex]angle \ DAB = 0.3(2x+10) = 90\\90/0.3 = 2x+10\\300 = 2x+10\\300-10=2x\\290=2x\\\\x=145[/tex]

Hence the answer is, x= 145

Answer:

29. 59.06°

30. 10.6

Step-by-step explanation:

29.
By using the Tangent angle rule, we can find the angle of elevation,

We know that

Tan Angle = opposite/adjacent

Tan x=AB/BC

Tan x=20/12

Tan x=5/3

[tex]x=Tan^{- }(\frac{5}{3})[/tex]

x=59.06°

30.

The law of sine is a formula that can be used to find the lengths of the sides of a triangle, or to find the angles of a triangle, when two sides and the angle between them are known. The formula is:

a / sin(A) = b / sin(B) = c / sin(C)

Here taking

a / sin(A) = c / sin(C)

here A=75°, C=180-75-50=55° and c -9 and

we need to find a,

substituting value

a/Sin(75°)=9/Sin(55°)

a=9*Sin(75°)/Sin(55°)

a=10.61

Therefore, the value of a is 10.6

Answer:

        tan (θ) = 20/12, and solve for θ

       We then get the equation tan (θ) =  20/12

        θ = arctan(5/3)

        Tan θ  =  opp/adj

        Tan θ  = 20/12

         θ  =  Tan^-1 (20/12)

          θ  =  59.03624346 degrees

           θ = 59.0 degrees

       Hence, the Angle of Elevation is ------->  59.0°

       m < C = 180 degrees - m<A - m<B

       m<C  = 180 degrees - 75 degrees  -  50 degrees

      m<C  =  55 degrees

       a/sin A  =  c/sin C

       a/sin 75 degrees  =  9/sin 55 degrees

        a  =  9 * sin 75 degrees/sin 55 degrees

        a  =  10.3

Therefore, The length of side A in --------> △ABC is approximately 10.3

Hope this helps you!

Answer:

a. Measure of skewness

Step-by-step explanation:

Skewness is a measure of the asymmetry of a probability distribution. It quantifies the extent to which a dataset's values deviate from a symmetric distribution. Various measures of skewness exist, including the Pearson's skewness coefficient, the Bowley skewness coefficient, and the moment coefficient of skewness. These measures provide a numerical indication of the skewness present in the dataset.

The area of the garden is 135 square yards.

Let's imagine you have a rectangular garden measuring 15 yards in length and 9 yards in width.

You want to find the area of this garden, which represents the amount of space inside the garden.

To find the area of a rectangle, you multiply the length by the width.

In this case, the length is 15 yards and the width is 9 yards.

Area = Length × Width

Area = 15 yards × 9 yards

Area = 135 square yards

This means that the garden can hold 135 square yards of grass, flowers, or any other objects you place inside it.

It's worth noting that the unit of measurement for area is always squared, such as square yards in this example.

This is because area is a two-dimensional measurement, representing the space within a flat surface.

By using the formula to calculate the area of a rectangle, you can easily determine the amount of space enclosed by any rectangular area when given the length and width.

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The elliptical ceiling is approximately 30 ft high at the center.

To find the height of the elliptical ceiling at the center, we can use the properties of an ellipse.

In this case, the two foci of the ellipse represent the positions where Juan and his friend are standing.

The distance between the two foci is 80 ft, and Juan is 20 ft away from the nearest wall.

This means that the sum of the distances from any point on the ellipse to the two foci is constant and equal to 80 + 20 = 100 ft.

Since Juan is standing at one focus and the distance to the nearest wall is given, we can determine the distance from Juan to the farthest wall by subtracting the distance to the nearest wall from the sum of the distances.

Distance from Juan to the farthest wall = 100 ft - 20 ft = 80 ft.

The height of the elliptical ceiling at the center is equal to half of the distance between the nearest and farthest walls.

Height of elliptical ceiling = (80 ft - 20 ft) / 2 = 60 ft / 2 = 30 ft.

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The output value f(3) in the functions f( x ) = 3x + 5, f( x ) =  [tex]\frac{1}{2}x^2-1.5[/tex] and f( x ) = [tex]\frac{3}{2x}[/tex] is 14, 3 and 1/2 respectively.

Given the functions in the question:

f( x ) = 3x + 5

f( x ) =  [tex]\frac{1}{2}x^2-1.5[/tex]

f( x ) = [tex]\frac{3}{2x}[/tex]

To evaluate each function at f(3), we simply replace the variable x with 3 and simplify.

a)

f( x ) = 3x + 5

Replace x with 3:

f( 3 ) = 3(3) + 5

f( 3 ) = 9 + 5

f( 3 ) = 14

b)

f( x ) =  [tex]\frac{1}{2}x^2-1.5[/tex]

Replace x with 3:

[tex]f(3) = \frac{1}{2}(3)^2 - 1.5\\\\f(3) = \frac{1}{2}(9) - 1.5\\\\f(3) = 4.5 - 1.5\\\\f(3) = 3[/tex]

b)

f( x ) = [tex]\frac{3}{2x}[/tex]

Replace x with 3:

[tex]f(3) = \frac{3}{2(3)} \\\\f(3) = \frac{3}{6} \\\\f(3) = \frac{1}{2}[/tex]

Therefore, the output value of f(3) is 1/2.

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The distance John can ride his bike in 12 minutes is approximately 1.6 miles.

To find out the distance John can ride his bike in 12 minutes, we can use the information given about his rate of riding.

We are told that John can ride his bike 4 miles in 30 minutes. This implies that his rate of riding is 4 miles per 30 minutes.

To calculate the distance John can ride in 12 minutes, we need to determine the proportion of time he is riding compared to the given rate.

We can set up a proportion to solve for the unknown distance:

(4 miles) / (30 minutes) = (x miles) / (12 minutes)

Cross-multiplying, we get:

30 minutes * x miles = 4 miles * 12 minutes

30x = 48

Now, we can solve for x by dividing both sides of the equation by 30:

x = 48 / 30

Simplifying the fraction, we have:

x = 8/5

So, John can ride his bike approximately 1.6 miles in 12 minutes, at his current rate.

Therefore, the distance John can ride his bike in 12 minutes is approximately 1.6 miles.

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The number we're looking for, which when added to 24/5 results in 9, is 21/5 or 4.2 in decimal form.

Let's solve the equation: 24/5 + x = 9, where x represents the unknown number we're trying to find.

To isolate x, we'll start by subtracting 24/5 from both sides of the equation:

x = 9 - 24/5

To add these two fractions, we need a common denominator. The denominator of 9 is 1, and the denominator of 24/5 is 5. To find a common denominator, we multiply 1 by 5:

x = (9 * 5)/5 - 24/5

This gives us:

x = 45/5 - 24/5

Now we can combine the fractions with the same denominator:

x = (45 - 24)/5

Simplifying the numerator:

x = 21/5

Therefore, the number we're looking for is 21/5. In decimal form, it can be written as 4.2.

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Answer:

x = 8

Step-by-step explanation:

The given diagram shows a triangle with the length of two sides and its included angle.

To find the value of the missing variable x, we can use the Law of Cosines.

[tex]\boxed{\begin{minipage}{6 cm}\underline{Law of Cosines} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]

From inspection of the given triangle:

Substitute the values into the formula and solve for x:

[tex]\begin{aligned}x^2&=18^2+21^2-2(18)(21)\cos 22^{\circ}\\x^2&=324+441-756\cos 22^{\circ}\\x^2&=765-756\cos 22^{\circ}\\x&=\sqrt{765-756\cos 22^{\circ}}\\x&=8.00306228...\\x&=8\end{aligned}[/tex]

Therefore, the value of the missing variable x is x = 8, rounded to the nearest hundredth.