A5 has a value of -11. This can also be confirmed by looking at the words in order: 5, 1, -3, -7, and -11.
Using the given formula, we can find the value of the nth term by recursively applying the formula, starting from the first term:
a1 = 5
a2 = a1 - 4 = 1
a3 = a2 - 4 = -3
a4 = a3 - 4 = -7
a5 = a4 - 4 = -11
Therefore, the value of a5 is -11. We can also verify this by checking the sequence of terms: 5, 1, -3, -7, -11.
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Peace's average mark on her 5 maths tests was 88. If her lowest score was dropped, her new average would be 90. What is her lowest mark?
Her average score on the four remaining tests is indeed 90, which confirms that her lowest score was 80.
Let's assume Peace's lowest score on the five math tests is x.
According to the problem statement, her average mark on all five tests is 88. This means that the sum of her scores on all five tests is:
5 * 88 = 440
If her lowest score was dropped, then the sum of her scores on the remaining four tests would be:
4 * 90 = 360
We know that the sum of her scores on all five tests is 440, so we can write an equation:
440 - x = 360
Solving for x, we get:
x = 80
Therefore, her lowest mark was 80. We can check this by finding her average score after dropping her lowest score:
(88 + 88 + 88 + 88 + 90) / 5 = 90
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HELP ME PLEASE!!!! THIS IS DUE TODAY!!!
Answer:
Yes, the triangles are similar.
Step-by-step explanation:
The triangles are similar because of the three angle measurements.
1) One of the measurements are labeled, and the other two are implied.
2) The measurement of angles HMG and JMK are equal because they are vertical angles.
3) Lastly, because two of the angles are the same, we know that the last angles, GJK and HGJ are the same because the three angles must have a sum of 180°.
DW
Problem 2: Given are a segment AB and a ray CD. Use compass, straightedge, and pencil,
to construct a point X on CD such that
CX = 2 1/2 AB
m is parallel to AB, and passing through points C.
Therefore, m II AB.
In geometry, a line segment is a part of a line bounded by two distinct endpoints and contains every point of the line between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed segment includes two endpoints, an open segment does not include both endpoints; a half-open segment has exactly one end. In geometry, a line segment is usually represented using a line above the two end symbols (like AB).
We use the basic rules of construction to draw lines according to the problem.
Draw a straight line AB and take a point C outside of it. Using a ruler and compass to draw the AB line, follow the steps below:
Draw a line AB, and take a point C outside this line. Take any point P on AB.Connect C to P.Take P as the center of the circle, take a suitable radius, draw an arc and cut AB in D, and PC in E.With C as the center and the same radius as in the previous step, draw an arc FG, and PC to H.Adjust the compass to the length of DE. Without changing the compass aperture, draw an arc HG at point I with H as the center.Draw a line l connecting points C and I as shown in the figure.Therefore, the line l is parallel to the line AB.
Complete Question:
Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB using ruler and compasses only.
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The area of cross-section of a solid cylinder is 803. 84 ft2, and the height of the solid is 12. 25 ft. Find the volume of the solid
cylinder using the Cavalieri's Principle.
9800. 25 ft
9587. 15 f3
9847. 04 ft3
9807. 50 ft
V = π(√(803.84/π))^2(12.25) = 9587.15 ft^3. Hence, following Cavalieri's theory, the solid cylinder has a volume of around 9587.15 ft3.
According to Cavalieri's principle, if two solids have the same height and every plane section passing through one solid along the height has the same area as the corresponding plane section passing through the other solid, the two solids must also have the same volume.
A circle of radius r has the same cross-section as a solid cylinder. A = r2 calculates the circle's area.
Therefore, πr^2 = 803.84 r^2 = 803.84/π\sr = √(803.84/π)
The solid cylinder is 12.25 feet tall.
We may get the volume of the solid cylinder using the volume of a cylinder formula, V = r2h: V = π(√(803.84/π))^2(12.25) = 9587.15 ft^3
Hence, following Cavalieri's theory, the solid cylinder has a volume of around 9587.15 ft3.
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Write a formula for the number of seconds, in any number
of minute
Number of seconds = Number of minutes x 60
What is a minute?A minute is a unit of time measurement that is equal to 60 seconds. It is commonly used to measure short periods of time, such as the duration of a conversation or the time it takes to complete a task.
What is time measurement?Time measurement is the process of quantifying the duration of an event or the interval between two events. It involves measuring the elapsed time between a start point and an end point, or between two points in time.
To convert any number of minutes into seconds, we can use the formula:
Number of seconds = Number of minutes x 60
This formula works because there are 60 seconds in one minute. So, to convert minutes to seconds, we simply multiply the number of minutes by 60 to get the equivalent number of seconds.
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Tom will run at least 30 miles this week. So far, he has run 22 miles. What are the possible numbers of additional miles he will run?
Use t for the number of additional miles he will run.
Write your answer as an inequality solved for t .
pls put ur answer written pls help me this is so hard I'm crying
The calculated number of additional miles he would run must not exceed 8 miles
How many additional miles should he run?The following are symbols for inequalities and their corresponding meanings:
> signifies a value that is larger than another value< signifies a value that is smaller than another value≥ signifies a value that is larger than or equal to another value≤ signifies a value that is smaller than or equal to another value.The inequality format that expresses the information in the question is as follows:
Tom's minimum required running distance must be greater than or equal to the sum of the distance he has already run and the distance he plans to run.
Using this inequality, we can derive the equation
30 ≥ 22 + t
Where t represents the distance Tom plans to run.
To determine the value of t, we must simplify the equation by combining similar terms.
We can subtract 22 from both sides, resulting in
8 ≥ t.
Therefore, the maximum distance Tom plans to run is 8 miles.
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11. F(x) = 2x2² - 12x +9
a. Concave
Narrower / Wider / Same
b. Axis of Symm.
(Show work here. )
C. Vertex
(Show work here. )
Width: (Circle One)
d. Domain:
_/2
10637910-y
Range:
/2
Inlo4 lemobib A
-/2
$
T
-3
t
m
$43
N
P
3
-5
e. Y-intercept & Reflection:
n
Additional Point & Reflection: _______
(Show work here. )
C
/3
/2
a. To establish whether the function is concave up or concave down, we must obtain the function's second derivative. F(x) = 2x2 - 12x + 9 F'(x) = 4x - 12 F"(x) = 4x - 12 F"(x) = 4.
The function is concave up because the second derivative has a constant value of 4, which is positive. Since the coefficient of x2 is more than one, the function is narrower than the conventional parabola y = x2. b. The equation x = -b/2a gives the parabola's axis of symmetry, where a and b are the coefficients of the x2 and x terms, respectively.A = 2 and b = -12 for the function F(x) = 2x2 - 12x + 9. As a result, the axis of symmetry is x = -12/(2*2) = 3. c. The point at which The coordinates (h, k) define the parabola, where h = -b/2a and k = f. (h). We already determined that the axis of symmetry for the function F(x) = 2x2 - 12x + 9 is x = 3. So, h = 3. To find k, enter h = 3 into the function: F(3) = 2(3)^2 - 12(3) + 9 = -9 As a result, the vertex is (3, -9). Width: The parabola's width is the distance between the x-intercepts, which may be calculated using the quadratic formula.
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During the summer, every student became 5% taller. Eric was x before summer, after summer he was 151.2 cm.
Let's assume Eric's height before summer is x cm. After summer, he became 5% taller, which means his new height is 1.05x cm. We also know that his new height is 151.2 cm. So we can set up an equation:
1.05x = 151.2
To solve for x, we can divide both sides by 1.05:
x = 151.2 / 1.05
x = 144 cm
Therefore, Eric's height before summer was 144 cm.
Solve the system of equations algebraically. Verify your answer using the graph.
y = 4x – 5
y = –3
What is the solution to the system of equations?
((StartFraction one-fourth EndFraction, negative 3), –3)
((StartFraction one-half EndFraction, negative 3), –3)
(–3, (negative 3, StartFraction 2 over 3 EndFraction))Solve the system of equations algebraically. Verify your answer using the graph.
y = 4x – 5
y = –3
What is the solution to the system of equations?
((StartFraction one-fourth EndFraction, negative 3), –3)
((StartFraction one-half EndFraction, negative 3), –3)
(–3, (negative 3, StartFraction 2 over 3 EndFraction))
Answer:
Step-by-step explanation:
To solve the system of equations algebraically, we need to substitute the second equation into the first equation and solve for x:
y = 4x – 5 (equation 1)
y = -3 (equation 2)
Substitute equation 2 into equation 1:
-3 = 4x – 5
Add 5 to both sides:
2 = 4x
Divide both sides by 4:
x = 1/2
Now, we can substitute this value of x back into either equation to find y:
y = 4(1/2) – 5 = -3
Therefore, the solution to the system of equations is (1/2, -3).
To verify this solution using the graph, we can plot the two equations on the same set of axes:
y = 4x – 5 (red line)
y = -3 (blue line)
The two lines intersect at the point (1/2, -3), confirming our solution.
Therefore, the answer is (B) (1/2, -3).
Write the equation of a circle given a center of (-5,-12) and a radius of 7
Answer:
(x+5)^2 + (y+12)^2 = 49
Step-by-step explanation:
(x-h)^2 + (y-k)^2 = r ^2 is the general equation for circles, where r is the radius and h,k are the center's coordinates.
For what ranges of values, holding all else constant, could each of the objective function coefficients be changed without changing the optimal solution? (
The optimal solution of an objective function does not change when its coefficients are changed within a certain range of values. This range of values for each coefficient depends on the function itself and the value of other coefficients.
For example, if the objective function is 3x + 2y, then the range of values for the coefficient of x (3) can be changed from 2 to 4 without changing the optimal solution. Similarly, the coefficient of y (2) can be changed from 1 to 3 without changing the optimal solution.
Therefore, the ranges of values for each coefficient of the objective function, holding all else constant, can be changed without changing the optimal solution.
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A Level III video editor takes 20 hours to add special effects to a movie. A Level II video editor takes 26 hours and a Level I video editor takes 32 hours. Find how long it would take them to add the special effects if they all work together.
It would take them about 8.4 hours to add the special effects if they all work together.
What is formula for work done?1/T = 1/T1 + 1/T2 + 1/T3 +... + 1/Tn, where T is the time it takes for all n persons to complete the task and T1, T2,..., Tn are the times it takes for each person to do the task individually, is the formula for addressing issues involving work done by several people. This equation is based on the premise that each person's contribution to the activity is inversely proportionate to the time it takes them to do it. In other words, a person contributes more effort to the entire task the faster they work. This method may be used to determine how long it will take all n persons working together to do the assignment in total.
The work done is calculated by:
1/T = 1/T1 + 1/T2 + 1/T3
Here, we have T1 = 20, T2 = 26 and T3 = 32 hours for work done separately.
Substituting the values:
1/T = 1/20 + 1/26 + 1/32
1/T = (32/640) + (24/640) + (20/640)
1/T = 76/640
T = 640/76
T ≈ 8.4 hours
Therefore, it would take them about 8.4 hours to add the special effects if they all work together.
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There are 6000 registered voters in Roseville and 3/4 of these voters are registered Democrats. A survey indicates that 2/3 of the registered Democrats are in favor of Bond Measure A and 3/5 of the other registered voters are in favor of this measure.
The total number of voters in favor of Bond Measure A is 3900 voters in favor of Bond Measure A.
How to find out how many voters are in favor of Bond Measure A.First we need to calculate the total number of voters who are registered Democrats and the total number of voters who are not registered Democrats, and then calculate the number of voters in favor of the measure for each group and add them together.
Total number of registered Democrats:
3/4 of 6000 registered voters = 4500 registered Democrats
Number of registered Democrats in favor of Bond Measure A:
2/3 of 4500 registered Democrats = 3000 registered Democrats in favor of the measure
Number of registered Democrats not in favor of Bond Measure A:
4500 - 3000 = 1500 registered Democrats not in favor of the measure
Total number of voters who are not registered Democrats:
6000 - 4500 = 1500 voters who are not registered Democrats
Number of voters who are not registered Democrats and in favor of Bond Measure A:
3/5 of 1500 voters who are not registered Democrats = 900 voters who are not registered Democrats and in favor of the measure
Therefore, the total number of voters in favor of Bond Measure A is:
3000 registered Democrats in favor + 900 other registered voters in favor = 3900 voters in favor of Bond Measure A.
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very confused please help explan the answer
Answer:
[tex]a_{n}[/tex] = 3 + 5(n - 1)
Step-by-step explanation:
there is a common difference between consecutive terms , that is
8 - 3 = 13 - 8 = 18 - 13 = 5
this indicates the sequence is arithmetic with explicit formula
[tex]a_{n}[/tex] = a₁ + d(n - 1)
where a₁ is the first term and d the common difference
here a₁ = 3 and d = 5 , then
[tex]a_{n}[/tex] = 3 + 5(n - 1) ← is the explicit formula
For sample proportion 0.450 and sample size 31, what is the
standard error of the distribution of sample proportions? Enter
your answer with 3 decimal places.
The standard error of the distribution of sample proportions is approximately 0.090.
The formula to find the standard error of the distribution of sample proportions is given by:Standard error = √(p(1-p)/n)Where,p = Sample proportionn = Sample sizeGiven that the sample proportion is 0.450 and the sample size is 31, we can substitute the values into the formula to find the standard error.Standard error = √(0.450(1-0.450)/31) ≈ 0.090So, the standard error of the distribution of sample proportions is approximately 0.090.
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Which equation would help find the measure of ∠m? x 52 78 = 180 52 x = 180 78 180 52 = 78 x 180 x = 78 52
Answer:
A
Step-by-step explanation:
how to find slope of x + y = 9
Answer:
m = -1
Step-by-step explanation:
Let's change the equation to y = mx + b
m = the slope
x + y = 9
Subtract x both sides
y = -x + 9
m = -1
So, the slope is -1
Classify the quadrilateral by its most specific name. Explain your reasoning.
To classify a quadrilateral by its most specific name, we need to examine its sides and angles.
What is a quadrilateral?A quadrilateral is four-sided polygon with four angles. It is closed shape with straight lines that do not cross each other.
Here are the most common types of quadrilaterals and their defining characteristics:
Parallelogram - quadrilateral consist of two pairs of parallel sides.
Rectangle - parallelogram with four right angles.
Rhombus - parallelogram with four congruent sides.
Square - rectangle and a rhombus, with four right angles and four congruent sides.
Let's consider an example.Suppose we are given a quadrilateral with four sides of equal length and two pairs of opposite parallel sides. This means that it has the characteristics of a parallelogram and a rhombus.
Reasoning: The quadrilateral has four equal sides, which makes it a rhombus. Additionally, it has two pairs of parallel sides, which satisfies the criteria for a parallelogram. However, since a rhombus is a type of parallelogram with additional properties, it is the most specific name for this quadrilateral.
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A microwave was originally sold for $137 and has been marked down to $66. What is the percentage decrease for the microwave? ____%
The percentage decrease in the price of the microwave is 51.82%.
What is the percentage decrease?Percentage is used to determine the relative value of a digit as a number out of a hundred. The sign that is used to represent percentages is %. In order to express a value as a percentage, multiply the number by 100. Percentage is a measure of frequency.
Percentage decrease = (change in price / initial price) x 100
Change in price = initial price - new price
$137 - $66 = $71
(71/137) x 100 = 51.82%
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I’ll give brainlyyyyyy
(a) Compared to the graph of y = x, the graph of y = x - 4 is shifted downwards by 4 units.
Compared to the graph of y = x, the graph of y = x - 4 intersects the y-axis at the point (0, -4).
(b) Compared to the graph of y = x, the graph of y = 5/2 * x is steeper (has a greater slope).
Compared to the graph of y = x, the graph of y = 5/2 * x intersects the y-axis at the point (0, 0).
How to complete the statement(a)
Compared to the graph of y = x, the graph of y = x - 4 is shifted downwards by 4 units. The -4 in the second equation means shifting downward 4 units
Compared to the graph of y = x, the graph of y = x - 4 intersects the y-axis at the point (0, -4). hence the y intercept is -4
(b)
Compared to the graph of y = x, the graph of y = 5/2 * x is steeper (has a greater slope).
Compared to the graph of y = x, the graph of y = 5/2 * x intersects the y-axis at the point (0, 0). In this case both graphs intercepts the y-axis at same point which is at the origin
The graphs are attached
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25 POINTS
Where do the graphs of f of x equals cosine of the quantity x over 2 and g of x equals square root of 3 minus cosine of the quantity x over 2 intersect on the interval [0, 360°)?
300°
150° and 210°
60°
30° and 330°
The graphs of f of x equals cosine of the quantity x over 2 and g of x equals square root of 3 minus cosine of the quantity x over 2 intersect on the interval [0, 360°) at: D. 30° and 330°.
What is a point of intersection?In Mathematics and Geometry, a point of intersection simply refers to the location on a graph where two (2) lines intersect or cross each other, which is primarily represented as an ordered pair containing the point that corresponds to the x-coordinate (x-axis) and y-coordinate (y-axis) on a cartesian coordinate.
In this context, we can logically deduce that the point of intersection of the graphs would be at function f(x) = function g(x):
[tex]cos(\frac{x}{2} )=\sqrt{3} -cos(\frac{x}{2} )\\\\\sqrt{3}=cos(\frac{x}{2} ) + cos(\frac{x}{2} )\\\\\sqrt{3}=2cos(\frac{x}{2} )\\\\cos(\frac{x}{2} )=\frac{\sqrt{3}}{2} \\\\cos(\frac{x}{2} )=\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{\sqrt{3}} \\\\cos(\frac{x}{2} )= \frac{3}{2\sqrt{3}}[/tex]
x/2 = 15
x = 15(2)
x = 30
x/2 = 165
x = 2(165)
x = 330.
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Lanny got a short term job selling computers. He is paid on
commission. In order to impress customers, he bought a few nice suits. If he has
$20 000 in sales, he will lose $140. If he has$30 000 in sales, he will make a $90
profit. Determine the:
a. rate of change and initial value
b. the amount he needs to sell to break even
c. The amount he needs to sell in order to make $1000 profit
The rate of change and initial value are 0.023 and -600 respectively.
The amount he needs to sell to break even is $26,086.96.
The amount he needs to sell in order to make $1000 profit is $69,565.22.
What is the slope-intercept form?Mathematically, the slope-intercept form of the equation of a straight line is given by this mathematical expression;
y = mx + c
Where:
m represents the slope or rate of change.x and y are the points.c represents the y-intercept or initial value.Based on the information provided, a system of equations that models the situation is given by;
-140 = 20,000x + c ...equation 1.
90 = 10,000x + c ...equation 2.
By subtracting equation 1 from equation 2, we have:
230 = 10,000x
x = 0.023
For the initial value, we have:
90 = 10,000x + c
c = -140 - 20,000(0.023)
c = -600.
In order to break even, y must be equal to 0;
y = 0.023x - 600
0 = 0.023x - 600
x = 600/0.023
x = $26,086.96.
In order to make $1000 profit, Lanny must sell the following:
y = 0.023x - 600
1,000 = 0.023x - 600
x = 400/0.023
x = $69,565.22.
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Express 0939 as a fraction.
According the given question 0.0939 as a Fraction equals 939/10000.
What is fractiοn?Fractiοns are referred tο as a whοle's cοnstituent parts in mathematics. A single item οr a cοllectiοn οf items can be the cοmplete. When we really cut a slice οf cake frοm the cοmplete cake, the cοmpοnent represents the percentage οf the cake. The wοrd "fractiοn" cοmes frοm Latin.
To write 0.09390 as a fraction you have to write .09390 as numerator and put 1 as the denominator. Now you multiply numerator and denominator by 10 as long as you get in numerator the whole number.
.09390 = .09390/1 = 0.939/10 = 9.39/100 = 93.9/1000 = 939/10000
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Complete Question:
Express 0.09390.0939 as a fraction.
8) Is \( f(x)=6 x^{2}-\frac{4}{x} \) odd, even or neither? You must show all of your algebra. 10 points 9) Find the equation of the line that goes though \( (-1,3) \) and \( (2,-4) \). 10 points
This function is neither odd nor even. To show this, we can take the algebraic approach.
The definition of an odd function is that for every \( x \), \( f(-x) = -f(x) \).
Therefore, we can substitute \( -x \) for \( x \) and see if this statement is true.
\( f(-x) = 6(-x)^{2}-\frac{4}{-x} \)
\( f(-x) = -6 x^{2}+\frac{4}{x} \)
Since this is not equal to \( -f(x) \), which is \( -6 x^{2}-\frac{4}{x} \), this function is neither odd nor even.
9) The equation of the line that goes through the two points \( (-1,3) \) and \( (2,-4) \) is \( y = \frac{1}{3}x + \frac{17}{3} \).
To find the equation, we can use the point-slope formula. The slope of the line is \( \frac{-7}{3} \), which we can find using the two points.
\( m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{-4-3}{2-(-1)} = \frac{-7}{3} \)
The point-slope formula for the equation of the line is \( y - y_{1} = m(x - x_{1}) \). We can substitute in the slope and the point \( (-1,3) \) to get the equation.
\( y - 3 = \frac{-7}{3}(x - (-1)) \)
\( y - 3 = \frac{-7}{3}(x + 1) \)
\( y = \frac{-7}{3}x - \frac{7}{3} + 3 \)
\( y = \frac{1}{3}x + \frac{17}{3} \)
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PLEASE SOMEBODY HELP: Generate a symbolic rule for locating the point that divides a line segment into two parts so that the ratio of the lengths is m: n, with the point closer to the left endpoint.
Step-by-step explanation:
Let the coordinates of the left endpoint of the line segment be (x1, y1) and the coordinates of the right endpoint be (x2, y2). Let the point dividing the line segment be (x, y). Then the distance between the left endpoint and (x, y) is mx/(m+n) and the distance between (x, y) and the right endpoint is nx/(m+n).
Using the distance formula, we have:
[tex]distance \: between \: (x1, y1) \: and \: (x, y) = \sqrt{((x - x1)^2 + (y - y1)^2)} = \frac{mx}{(m+n)} [/tex]
[tex]distance \: between \: (x, y) \: and \: (x2, y2) = \sqrt{((x2 - x)^2 + (y2 - y)^2)} = \frac{nx}{(m+n)} [/tex]
Squaring both equations, we get:
[tex](x - x1)^2 + (y - y1)^2 = \frac{mx}{(m+n)^2}[/tex]
[tex](x2 - x)^2 + (y2 - y)^2 = \frac{nx}{(m+n)^2}[/tex]
Expanding the squares, we get:
[tex]x^2 - 2x1x + x1^2 + y^2 - 2y1y + y1^2 = \frac{mx}{(m+n)^2} [/tex]
[tex]x^2 - 2x2x + x2^2 + y^2 - 2y2y + y2^2 = \frac{nx}{(m+n)^2}[/tex]
Rearranging and simplifying, we get:
[tex]x = \frac{(mx2 + nx1)}{(m+n)} \: and \: y = \frac{(my2 + ny1)}{(m+n)}[/tex]
Therefore, the point that divides the line segment into two parts so that the ratio of the lengths is m: n and the point is closer to the left endpoint is:
[tex](x, y) = \frac{(mx2 + nx1)}{(m+n)}, \frac{(my2 + ny1)}{(m+n)}[/tex]
Billy ran 52 yards. How many feet did he run?
Answer:
The answer to your problem is, 156
First lets find out how many feet or in one yard ?
Well 1 yard = 3 feet.
Now that we know that we can use our knowledge to answer the next question:
If Bill ran 52 yards, how many feet did he ran?
We can do math to solve our answer. :
1 x 52 = 52.
Now we need to multiply 52 x 3.
That is a little bit to much, so lets make the problem easier!
By breaking up the problem \/ to make it more easier.
50 x 3. ( 5 x 3 = 15, just add a zero! ) = 150
3 x 2 = 6.
Next add:
150 + 6 = 156.
Thus the answer to your problem is, 156
The daily cost of production in a factory is calculated using ()= 200+ 16, where is the number of complete products manufactured. Which set of numbers best defines the domain of ()?
Integers
positive real numbers
positive rational numbers
whole numbers
The domain of the function is positive integers. The function () = 200 + 16 represents the daily cost of production in a factory, where x is the number of complete products manufactured.
Since the production of products cannot be negative or fractional, the domain of the function is restricted to non-negative integers.
Therefore, the domain of the function is the set of positive integers:
{1, 2, 3, 4, ...}
Any value outside this set would not make sense in the context of the problem, as a fraction or a negative number of products cannot be produced.
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What single percentage change is equivalent to a 11% increase followed by a 21% decrease?
Answer:
Step-by-step explanation:
-10 perecent
because 21-11=10
Find the equation of a line that passes through the points (1,-3) and (3,-4).
[tex](\stackrel{x_1}{1}~,~\stackrel{y_1}{-3})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{-4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-4}-\stackrel{y1}{(-3)}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{1}}} \implies \cfrac{-4 +3}{2} \implies \cfrac{ -1 }{ 2 } \implies - \cfrac{ 1 }{ 2 }[/tex]
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-3)}=\stackrel{m}{- \cfrac{ 1 }{ 2 }}(x-\stackrel{x_1}{1}) \implies y +3 = - \cfrac{ 1 }{ 2 } ( x -1) \\\\\\ y+3=- \cfrac{ 1 }{ 2 }x+\cfrac{1}{2}\implies y=- \cfrac{ 1 }{ 2 }x+\cfrac{1}{2}-3\implies {\Large \begin{array}{llll} y=- \cfrac{ 1 }{ 2 }x-\cfrac{5}{2} \end{array}}[/tex]
Answer: First, let's find the slope of the line:
slope = (change in y) / (change in x)
slope = (-4 - (-3)) / (3 - 1)
slope = -1/2
Now, let's choose one of the points, say (1,-3), and use the point-slope formula:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) are the coordinates of the point.
So, substituting in the values we get:
y - (-3) = (-1/2)(x - 1)
Simplifying this equation, we get:
y + 3 = (-1/2)x + 1/2
Subtracting 3 from both sides, we get:
y = (-1/2)x - 5/2
Therefore, the equation of the line that passes through the points (1,-3) and (3,-4) is y = (-1/2)x - 5/2.
Your welcome.
Step-by-step explanation:
A) Given: Regular pyramid KO ⊥ (ABCD) KO = 10. 2, m∠AKO = 38º Find: V
The standard pyramid KO has a volume of roughly 64.88 cubic units.
To find the volume of the given regular pyramid KO, we need to know the area of the base and the height of the pyramid. Since the pyramid is regular, the base is a square. Let the side of the square base be x.
Now, we can use trigonometry to find the height of the pyramid. Draw a line from vertex K to the center O of the base. This line is perpendicular to the base, so it splits the base into two equal parts. Let M be the midpoint of side AB. Then, we have right triangle KMO with hypotenuse KO, and angle MKO equal to half of angle AKO (19 degrees).
We can use the trigonometric function tangent to find the height h of the pyramid:
tan 19 = h/5 (where 5 is half of x, the side length of the base)
h = 5 tan 19 ≈ 1.98
Now, we can use the formula for the volume of a pyramid:
V = (1/3) * area of base * height
Since the base is a square, its area is x². Substituting the known values, we get:
V = (1/3) * x² * h
V = (1/3) * x² * 1.98
V = (0.66) x²
We are given that KO = 10.2, which is the slant height of the pyramid. Using the Pythagorean theorem, we can find x:
x² + (5/2)² = 10.2²
x² = 104.04 - 6.25
x² = 97.79
x ≈ 9.89
Substituting this value for x into the equation for the volume, we get:
V = (0.66) * 9.89²
V ≈ 64.88
Therefore, the volume of the regular pyramid KO is approximately 64.88 cubic units.
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