What are the coordinates of the point on the directed line segment from ( − 7 , 3 ) (−7,3) to ( 3 , − 9 ) (3,−9) that partitions the segment into a ratio of 3 to 1?

An equation for the function graphed above is [tex]y = \frac{4(x - 1)}{(x + 1)(x - 4)}[/tex].

In Mathematics and Geometry, the vertical asymptote of a function simply refers to the value of x (x-value) which makes its denominator equal to zero (0).

By critically observing the graph of this rational function shown above, we can logically deduce that its vertical asymptotes include the following;

x = -1 ⇒ x + 1 = 0.

x = 4 ⇒ x - 4 = 0.

In this context, an equation that represent its denominator is given by the following quadratic function:

Q(x) = (x + 1)(x - 4)

Assuming the leading coefficient of the function P(x) is represented by a;

P(x) = a(x - 1)

By evaluating and solving for the leading coefficient "a" in this function, we have;

f(x) = P(x)/Q(x)

f(x) = a(x - 1)/[(x + 1)(x - 4)]

f(0) = a(0 - 1)/[(0 + 1)(0 - 4)]

f(0) = -a/[(1)(-4)]

f(0) = a/4

Since the y-intercept of this graph is located at (0, 1), the value of the leading coefficient (a) can be calculated as follows;

1 = a/4

1 = a/4

a = 4 × 1

a = 4

By substituting the value of the leading coefficient (a) into the function, we have the following;

f(x) = P(x)/Q(x)

f(x) = y = 4(x - 1)/[(x + 1)(x - 4)]

[tex]f(x) = y = \frac{4(x - 1)}{(x + 1)(x - 4)}[/tex]

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Answer: y = [tex]\frac{3}{2}[/tex]x - 12

Step-by-step explanation:

     First, we will find the slope of the equation. The x-intercept means that when x = 8, y = 0 giving us the point (8, 0). The y-intercept means that when y = -12, x = 0, giving us the point (0 -12).

     Now, we will find change in y over change in x (the slope).

               [tex]\displaystyle \frac{-12-0}{0-8}=\frac{-12}{-8} =\frac{3}{2}[/tex]

     Lastly, we can write our equation using the slope and y-intercept.

               y = [tex]\frac{3}{2}[/tex]x - 12

Answer:

y = 3/2x - 12

Step-by-step explanation:

X-intercept located at (8,0)

Y-intercept located at (0, -12)

The equation is y = mx + b

m = the slope

b = y-intercept

Slope = rise/run or (y2 - y1) / (x2 - x1)

2 points (8,0) (0, -12)

We see the y decrease by 12 and the x decrease by 8, so the slope is

m = -12 / -8 = 3/2

Y-intercept is = -12

So, the equation is y = 3/2x - 12

The distance made good by the ship is approximately 259.5 nautical miles.

What is distance?

Distance is a measurement of how far apart two objects or points are, either numerically or rarely qualitatively. Distance can relate to a physical length in physics or to an estimate based on other factors in everyday language.

To find the distance made good, we need to calculate the total displacement of the ship. This can be found using the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

In this case, the 120 nm southward and 230 nm westward movements of the ship form the legs of a right triangle, and the hypotenuse represents the total displacement. Using the Pythagorean theorem, we get:

Total displacement = √((120²) + (230²)) nm

= √(14400 + 52900) nm

= √(67300) nm

≈ 259.5 nm

Therefore, the distance made good by the ship is approximately 259.5 nautical miles.

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The equation of the circle after it is dilated by a scale factor of 2 is given as follows:

B. (x + 6)² + (y - 4)² = 36.

A dilation can be defined as a transformation that multiplies the distance between every point in an object and a fixed point, called the center of dilation, by a constant factor called the scale factor.

The equation of the circle in this problem is given as follows:

(x + 3)² + (y - 2)² = 9.

The features are:

After the dilation by a scale factor of 2, the features are given as follows:

Then the equation of the circle would be of:

(x + 6)² + (y - 4)² = 36.

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Thus, the probability that either blue or green or purple will appear on the next spin is 63%.

The proportion between the number of likely occurrences and the total number of potential occurrences.

Given result for the spinner:

probability = favourable outcome / total outcome

probability (blue) = 4/35

probability (Purple) = 3/35

probability (Green) = 15/35

So, probability (either blue or green or purple) = 4/35 + 3/35 + 15/35

= 22/35

= 0.6285

= 62.85

= 63%

Thus, the probability that either blue or green or purple will appear on the next spin is 63% (nearest whole number).

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Answer:

a) The surface area of a cuboid is given by the formula:

SA = 2lw + 2lh + 2wh

where l, w, and h are the length, width, and height of the cuboid, respectively.

In this case, we are given that:

h = xem

w = 2h = 2xem

l = 3h = 3xem

Substituting these values into the formula for the surface area of a cuboid, we get:

SA = 2(3xem)(2xem) + 2(3xem)(xem) + 2(2xem)(xem)

= 12xem^2

Therefore, the formula for the surface area of this cuboid is SA = 12xem^2.

b) Using the formula SA = 12xem^2, we can calculate the surface area of each cuboid and then add them to find the total surface area.

For cuboid A, x = 3, so the surface area is:

SA(A) = 12(3em)^2

= 108em^2

For cuboid B, x = 5, so the surface area is:

SA(B) = 12(5em)^2

= 300em^2

Therefore, the total surface area of the two cuboids is:

SA(A+B) = SA(A) + SA(B)

= 108em^2 + 300em^2

= 408em^2

So the total surface area of the two cuboids is 408em^2.

The length of a rectangle cannot be negative, hence the length of the rectangle is 7 units.

Based on the question given, the width (W) is the length (L) minus 3 units, which can be expressed as:

W = L - 3

The formula for the area (A) of a rectangle is:

A = Length × Width

We are given that the area is 28 square units, so:

A = 28

Now, we can substitute the expressions for width (W) and area (A) into the area formula:

28 = L × (L - 3)

Now, we need to solve this quadratic equation for L. First, expand the equation:

28 = L^2 - 3L

Now, rearrange the equation in standard quadratic form

L^2 - 3L - 28 = 0

To solve this quadratic equation, you can factor it:

(L - 7)(L + 4) = 0

Now, set each factor equal to zero and solve for L:

L - 7 = 0

L = 7

L + 4 = 0

L = -4

Since the length of a rectangle cannot be negative, hence the length of the rectangle is 7 units.

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The correct response is 4. We can consider the following construction to understand why: We drew a 3 cm radio circle with the center located at P. Llamemoso C1.

Half of a circle's diameter corresponds to its radius. The radius equals 5 cm for a 10 cm diameter.

So, in order for C2 to intersect C, the straight line connecting P and C's center must be contained within C2, that is, it must be separated from C's center by a distance less than r. In other words, we need that:

r > d + 3

the distance between the center of C and the straight line connecting P and the center of C is where d. Its distance, d, is same to that between P and the center of C, minus C1's radio. Defined as,

d = PC - 3

where PC denotes the distance between P and C's center.

Thus, we require that:

r > PC - 3 + 3 = PC

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Answer:

62.37 in

Step-by-step explanation:

Hope this helped!

Answer:252

Step-by-step explanation:

14x18=252

The possible dimensions of the rectangular plot are 13 m by 13 m (a square) or 26 m by 6.5 m.

Perimeter is the total distance around the edge of a two-dimensional shape, such as a polygon or a circle. It is calculated by adding up the lengths of all the sides of the shape. The perimeter is a measure of the distance that must be covered to go around the shape, and is typically expressed in units of length, such as meters or feet.

Let the length of the rectangular plot be L and the width be W.

We know that the perimeter of the rectangular plot is equal to the sum of the lengths of the three sides of the fence:

2L + W = 39

Also, the area of the plot is LW = 169.

From the second equation, we can solve for L or W in terms of the other variable:

L = 169/W or W = 169/L.

Substitute these expressions into the first equation:

2(169/W) + W = 39

Simplify and rearrange:

2(169) + W² = 39W

W² - 39W + 338 = 0

Solve for W using the quadratic formula:

W = (39 ± √(39² - 4(1)(338))) / 2

W = (39 ± 13) / 2

W = 26 or 13.

If W = 26, then L = (39 - 2W)/2 = -6, which is not possible. So, we can discard this solution.

If W = 13, then L = (39 - 2W)/2 = 13.

Therefore, the possible dimensions of the rectangular plot are 13 m by 13 m (a square) or 26 m by 6.5 m.

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Therefore, the factored form of the expression [tex]-5x^3+10x^2-3x+6 is -(5x^2+3)(x-2).[/tex]

In mathematics, a factor refers to a number or an algebraic expression that is multiplied by another number or expression.

For example, in the expression 6 x 8 = 48, 6 and 8 are factors of 48 because they are both multiplied together to get the product 48. Similarly, in the algebraic expression 3x(x+2), 3x and (x+2) are both factors of the expression.

by the question.

Question 4:

To factor the expression. [tex]2x^5+4x^3-6x^2-12[/tex], we can first take out a common factor of 2 from all the terms:

[tex]2(x^5+2x^3-3x^2-6)[/tex]

Next, we can factor the expression in the parentheses by grouping the first two terms and the last two terms:

[tex]2(x^5+2x^3-3x^2-6) = 2(x^5-3x^2) + 2(2x^3-6)[/tex]

[tex]= 2x^2(x^3-3) + 4(x^3-3)= (2x^2+4)(x^3-3)[/tex]

Therefore, the factored form of the expression[tex]2x^5+4x^3-6x^2-12 is (2x^2+4)(x^3-3).[/tex]

Question 5:

To factor the expression [tex]3x^7-8x^6-9x^5+24x^4[/tex], we can first take out a common factor of [tex]3x^4[/tex] from all the terms:

[tex]3x^4(x^3-8x^2-3x+8)[/tex]

Next, we can factor the expression in the parentheses by grouping the first two terms and the last two terms:

[tex]3x^4(x^3-8x^2-3x+8) = 3x^4(x^2(x-8)-1(x-8))= 3x^4(x-1)(x-8)(x^2+1)[/tex]

Therefore, the factored form of the expression[tex]3x^7-8x^6-9x^5+24x^4 is 3x^4(x-1)(x-8)(x^2+1).[/tex]

Question 6:

To factor the expression. [tex]-5x^3+10x^2-3x+6[/tex], we can first take out a common factor of -1 from all the terms:

[tex]-1(5x^3-10x^2+3x-6)[/tex]

Next, we can factor the expression in the parentheses by grouping the first two terms and the last two terms:

[tex]-1(5x^3-10x^2+3x-6) = -1(5x^2(x-2)+3(x-2))= -(5x^2+3)(x-2)[/tex]

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The value of the missing length indicated is 128 units.

Two or more triangles are said to be similar when on comparing their properties, they have some common relations. But similarity is not congruency.

Let the missing length indicated be represented by x. So that comparing the triangles, we have;

21/ 77 = 48/ (48 + x)

21(48 + x) = 48*77

1008 + 21x = 3696

21x = 3696 - 1008

      = 2688

x = 2688/ 21

  = 128

x = 128

The missing length indicated is 128 units.

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To find how long it will take for Francine's money to grow to $7,500, we can set up the equation:

7500 = 5000(1.005)^(12t)

where t is the number of years. We want to solve for t.

First, divide both sides by 5000:

1.5 = (1.005)^(12t)

Next, take the natural logarithm of both sides:

ln(1.5) = ln(1.005)^(12t)

Using the rule of logarithms that ln(a^b) = b ln(a), we can simplify the right side:

ln(1.5) = 12t ln(1.005)

Finally, divide both sides by 12 ln(1.005):

t = ln(1.5)/(12 ln(1.005))

Using a calculator, we get:

t ≈ 14.6 years

Therefore, it will take about 14.6 years for Francine's money to grow to $7,500.

Answer:

If the radius of the pastry stamp Petra uses to stamp circles in the dough for the donuts is 4.25 inches, then the diameter is 2 * 4.25 = 8.5 inches. The area of each donut can be calculated using the formula for the area of a circle:

Area of a circle = π * radius^2

So, the area of each donut is:

π * (4.25)^2 = 56.75π square inches (rounded to two decimal places)

If the radius of the pastry stamp Petra uses to stamp out circles for the donut holes is 0.4 inches, then the diameter is 2 * 0.4 = 0.8 inches. The area of each donut hole can be calculated using the same formula:

Area of a circle = π * radius^2

So, the area of each donut hole is:

π * (0.4)^2 = 0.16π square inches (rounded to two decimal places)

Hope This Helps!

Answer:

Step-by-step explanation:

But we know that √5 is an irrational number. So, {(a - 3b)/b} should also be an irrational number. Hence, it is a contradiction to our assumption. Thus, 3 + √5 is an irrational number.

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The probability that 2 particular health professionals will be selected in the subcommittee is 4/15. There are 210 ways to select a subcommittee of 4 health professionals from the 10 total health professionals.

Probability acts as an indicator of how likely an event is to occur.

It is a number between 0 and 1, with 0 indicating an impossible event and 1 suggesting a certain event.

The number of favorable outcomes divided by the total number of possible outcomes yields the probability of an event.

The number of ways to select a subcommittee of 4 health professionals out of the 10 total health professionals is given by the combination formula:

C(10, 4) = 10! / (4! * 6!) = 210

To calculate the probability that 2 particular health professionals will be selected in the subcommittee, we need to consider the number of ways that we can choose the other 2 health professionals from the remaining 8. The number of such combinations is given by the combination formula:

C(8, 2) = 8! / (2! * 6!) = 28

So the probability that the subcommittee will consist of the 2 particular health professionals we are interested in, as well as 2 other health professionals, is:

(2 choose 2) * (8 choose 2) / (10 choose 4) = 1 * 28 / 210 = 4/15

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Anna needs to pay N$8,500 + N$1,062.50 = N$9,562.50 in two and a half years from now to meet her obligation.

What is Simple interest?

Simple interest is a type of interest that is calculated based only on the principal amount of a loan or investment. It is a fixed percentage of the principal that is charged for the use of money over a specified period of time.

First, let's calculate the interest on the loan for three and a half years.

Interest = Principal * Rate * Time

where the principal is N$20,000, the rate is 5%, and the time is 3.5 years.

Interest = 20,000 * 0.05 * 3.5 = N$3,500

So Anna owes N$20,000 + N$3,500 = N$23,500 in total at maturity.

Next, let's subtract the payments Anna plans to make from the total amount she owes to find out how much she still needs to pay:

N$23,500 - N$8,000 (planned payment) - N$2,000 (planned payment) - N$5,000 (planned payment) = N$8,500

So Anna still needs to pay N$8,500 to meet her obligation.

Finally, we need to calculate how much she should pay in two and a half years from now. We can use the simple interest formula again, this time with a principal of N$8,500, a rate of 5%, and a time of 2.5 years.

Interest = Principal * Rate * Time

Interest = 8,500 * 0.05 * 2.5 = N$1,062.50

So Anna needs to pay N$8,500 + N$1,062.50 = N$9,562.50 in two and a half years from now to meet her obligation.

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The functiοns are f(-3)=2,  f(1)=4 and f(4)= -2.

In mathematics, a functiοn is  a relatiοn between a set οf inputs having οne οutput each. We can explain that  a functiοn is a relatiοnship between inputs where each input is related tο exactly οne οutput. A functiοn is generally denοted by g(x) where x is the input. The general representatiοn οf a functiοn is y = g(x).

The given cοοrdinates are (-3,2); (1,4);(4,-2)

Frοm the given cοοrdinates the functiοns can be fοrmed.

Let the functiοn be y=f(x)

Frοm the given cοοrdinate system the first value is x- value and the secοnd value is y-cοοrdinate.

Sο cοnsidering the first cοοrdinate -3 is the x-cοοrdinate and 2 is the y cοοrdinate and similarly fοr οthers.

Hence the functiοn will be f(-3)=2

Similarly frοm the secοnd cοοrdinate the functiοn will be f(1)=4

[ As 1 is the x- cοοrdinate and 4 is the y-cοοrdinate]

Frοm the third cοοrdinate the functiοn will be f(4)= -2

Hence, the functiοns are

f(-3)=2

f(1)=4

f(4)= -2

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Thus, the value of 'x' for the centroid of triangle ABC is found as: x = 3.

The intersection of the triangle's three medians is known as the centroid. The segments that join a vertex to the opposing side's midway are known as medians.

The centroid for triangle ABC in this image is located at point D. It is simplest to create all three medians and seek for the point of intersection to determine a triangle's centroid. The centroid is the point at which gravity is greatest for a triangle formed of a uniform material.

From the property of centroid of triangle:

BD = 2*DF

Put the given values:

6x+6 = 2(8x - 12)

6x + 6 = 16x - 24

10x = 30

x = 3

Thus, the value of 'x' for the centroid of triangle ABC is found as: x = 3.

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Answer:

108.16 sq ft

Step-by-step explanation:

We know it is a square bedroom, meaning all sides are equal. So then we do 10.4*10.4 or 10.4^2 and get 108.16 sq ft

Based on the information, we can infer that the gumball machine had 530 blue gumballs, 106 red gumballs, 848 green gumballs, 424 pink gumballs, 212 yellow gumballs.

To find the amount of gum of each color that was in the machine we must perform rules of three as shown below:

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The fοur integers are all equal tο 9, and the range is 12.

The cοllectiοn, analysis, interpretatiοn, presentatiοn, and οrganisatiοn οf data are all tοpics that fall under the umbrella οf statistics, a subfield οf mathematics.

Let's call the fοur integers a, b, c, and d.

Since the median is 9, we knοw that either b οr c must be 9. Let's assume b = 9.

Since the mοde is 9, we knοw that either a, c, οr d must be 9. Let's assume a = 9.

Nοw we have twο cases:

Case 1: c = 9

In this case, we have a, b, c, and d all equal tο 9, which gives us a mean οf 9.

Case 2: d = 21

In this case, we have a = 9, b = 9, c = 12, and d = 21, which gives us a mean οf 12.

Hοwever, we were tοld that the mean is 9, sο we can cοnclude that the first case is the cοrrect οne.

Hence, the fοur integers are all equal tο 9, and the range is 12.

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Complete Question:

Four integers have a mean of 9, a median of 9, a mode of 9 and a range of 12.

Find the four integers

Answer:

Answer: C. 40°

Step-by-step explanation:

The shape in the picture is an isosceles triangle. This is determined because two sides share the same length. This will mean that two angles will also share the same measurement, helping us to find our answer.

Step 2: Write an equation

Now that we have identified the type of triangle, we can write an equation to solve. Two angles will be equal to 70, so subtract 70 twice from 180. It will be taken away from 180, since in a triangle the three angles are equal to 180 degrees. Now we can write the equation!

X= 180-70-70

Step 3: Solve for ∠x

The final step will be to solve using the equation we have created before.

X= 180-70-70

X= 110-70

X= 40

There is the final answer. Hope this helps! Comment below for more questions.

Answer:

Answer: 4,772.8 cm2

Step-by-step explanation:

The company will need 3,583.2 square inches of paper to make 15 labels.

What is cylinder?

Traditionally, a cylinder was a three-dimensional solid, one of the most fundamental curvilinear geometric shapes. It is a prism with a circle as its base in elementary geometry.

The formula for circumference is:

Circumference = π × diameter

Circumference = 3.14 × 8in

Circumference = 25.12in

So the label will need to be 25.12 inches long.

The height of the label will be the same as the height of the can, which is 9 inches.

To find the total area of the paper needed for 15 labels, we need to multiply the length of each label by its height and then multiply that by the number of labels:

Total area = length × height × number of labels

Total area = 25.12in × 9in × 15

Total area = 3,583.2 square inches

Thus, the answer is 3,583.2 square inches.

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The maximum daily profit the company can earn is $2,350.

In mathematics, a graph of a function is a visual representation of the relationship between the input and output values of the function.

It is a set of points in a coordinate plane that represents the values of the function for different inputs.

The function P(x) = −6x² + 180x + 1,000 models the daily profit of the company, where x is the number of $5 price increases for each solar panel. The graph of the function has a vertex at approximately (15, 2350), which represents the maximum point on the graph.

Therefore, the maximum daily profit the company can earn is $2,350.

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Answer:

Step-by-step explanation:

You want the angular speed in RPM and in radians per minute for a truck wheel 38 inches in diameter traveling at 60 mph.

There are 60 minutes in an hour, so a speed of 60 mi/h is a speed of ...

  [tex]\dfrac{60\text{ mi}}{1\text{ h}}\times\dfrac{1\text{ h}}{60\text{ min}}\times\dfrac{5280\text{ ft}}{1\text{ mi}}=5280\text{ ft/min}[/tex]

The distance traveled for 1 revolution of the wheel is equal to its circumference:

  C = πd

  C = π(38 in)/(12 in/ft) = (19/6)π ft ≈ 9.948 ft

Then the number of revolutions in 5280 ft is ...

  (5280 ft/min)/(9.948 ft/rev) ≈ 530.7 rev/min

The wheels make about 530.7 revolutions per minute.

Each revolution is an angle of 2π radians, so the angular speed is ...

  ω = 530.7 rev/min × 2π rad/rev

  ω = 3334.7 rad/min