6Aonly valid solution is 5π/4 on the interval [0, 2π]
63,cos x = sin x - 1 is x = 30°
To solve the following trigonometric equations, follow the steps given below:sec x + 1 = tan xcos x = sin x - 12π is the solution to the trigonometric equation sec x + 1 = tan x on the interval [0, 2π]. Let's try to solve the given equation:sec x + 1 = tan xLet's use the identity for secant and tangent in terms of cosine and sine:sec x = 1/cos xtan x = sin x / cos xSubstitute these into the given equation:1/cos x + 1 = sin x / cos xThe denominator of the right-hand side can be cleared by multiplying both sides by cos x. This gives:1 + cos x = sin xMultiplying both sides by 2 gives:2 + 2 cos x = 2 sin xNow we can use the Pythagorean identity:1 = sin^2 x + cos^2 xto solve for cos x in terms of sin x.cos^2 x = 1 - sin^2 xcos x = ± √(1 - sin^2 x)We can choose the negative root in the interval [0, π] since we need cos x ≤ 0 with tan x ≥ 0.Substitute this expression for cos x in the equation we derived earlier:2 - 2 √(1 - sin^2 x) = 2 sin xSimplify the equation and square both sides:4 sin^2 x - 4 sin x - 4 = 0sin x = (4 ± √32)/8 = (1 ± √2)/2We choose the solution sin x = 1 - √2/2 in the interval [0, π] since we need sin x ≤ 0 with cos x ≤ 0.Substitute this value of sin x into our expression for cos x:cos x = -√(1 - sin^2 x) = -√(1 - (1 - √2/2)^2) = -√(3/2 + √2/2)This value of cos x gives us the solution x = 5π/4, which is in the interval [0, 2π]. Therefore, the only valid solution is 5π/4 on the interval [0, 2π].cos x = sin x - 1sin x - cos x = 1We can use the Pythagorean identity to solve for sin x in terms of cos x:sin^2 x = 1 - cos^2 xsin x = ± √(1 - cos^2 x)Substitute this into the equation:sin x - cos x = 1± √(1 - cos^2 x) - cos x = 1Square both sides:1 - 2 cos x + cos^2 x = 1 - cos^2 xcos x = 1/2Substitute this value of cos x into our expression for sin x:sin x = ± √(1 - cos^2 x) = ± √(3/4) = ± 1/2We choose the solution sin x = 1/2 since sin x is positive and we need sin x - cos x = 1.Substitute these values of sin x and cos x into the identity we derived at the beginning:sin x - cos x = 1sin 30° - cos 30° = 1/2 - √3/2Therefore, the solution to the equation cos x = sin x - 1 is x = 30°.
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Start by creating two different circles:
`◍ Create a point, and label it A. To make the remainder of the activity easier, choose integers for the x and y coordinates of point A.
◍ Create a circle with its center at point A and with a radius of your choice. To make the remainder of the activity easier, choose an integer value for the radius.
◍ Create another point, and label it B. To make the remainder of the activity easier, choose integers for the x and y coordinates of the point.
◍ Create a circle with its center at point B and with a radius of your choice that is different from the radius chosen for circle A. To make the remainder of the activity easier, choose an integer value for the radius
The integers and radius for different circles as instructed are; (1.) Circle A (1, 3) with r= 3 (2.) Circle B (5, 3) with r= 5.
How to create the circles with integers and radius as instructed?1. On a y and x graph, create a dot and get the integers for the dot which are the numbers you find when you trace the dot to the x and y axis.
Integers are a set of numbers that include positive numbers, negative numbers, and zero. Integers are whole numbers (not fractions or decimals) that can be represented on the number line. Examples of integers include -3, -2, -1, 0, 1, 2, 3, and so on.
For the diagram below, diagram A and B share a similar point at the y axis and a different point at the x axis. There integers are
A ( 1, 3) and
B (5, 3)
The radius of choice for A is r= 3
The radius of choice for B is r= 5
Therefore the diagram should look like the one in the attached file.
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Calculate the difference between the amount of commission for selling goods worth Rs.20 lakhs and Rs .30 lakes. Sales between 15-25 lakhs is 1% and sales between 25-54 is 1.5%.
The difference in commission for selling goods worth Rs. 20 lakhs and Rs. 30 lakhs is Rs. 7,500.
How to calculate the difference between the amount of commission for selling goods ?
The commission on sales worth Rs. 20 lakhs would be:
1% of (20 lakhs) = 0.01 x 20,00,000 = Rs. 20,000
The commission on sales worth Rs. 30 lakhs would be:
1% of (25 lakhs) + 1.5% of (5 lakhs) = (0.01 x 25,00,000) + (0.015 x 5,00,000) = Rs. 27,500
The difference in commission between the two sales would be:
Rs. 27,500 - Rs. 20,000 = Rs. 7,500
Therefore, the difference in commission for selling goods worth Rs. 20 lakhs and Rs. 30 lakhs is Rs. 7,500.
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Calculate how far a ball would be hit if it was hit at an angle of 25° and a velocity of 80 mph.
The distance that the ball was hit, considering that it was hit at an angle of 25° and a velocity of 80 mph, is given as follows:
d = 99.7 meters.
How to obtain the distance that the ball was hit?First we must obtain the time that the ball was in the air, and the equation is given as follows:
t = 2vy/g.
In which:
vy is the y-component of the velocity.g is the gravity.The y-component of the velocity is given as follows:
vy = vsin(y)
vy = 35.71 x sine of 25 degrees. (35.71 m/s = 80 mph)
vy = 15.1 m/s.
As the gravity is of 9.81 m/s², the time is given as follows:
t = 2 x 15.1/9.81
t = 3.08 s.
The distance is then given as follows:
d = Vxt.
d = 35.71 x cosine of 25 degrees x 3.08
d = 99.7 meters
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Examine the solution process below. If it is correct, select "The solution is correct." If incorrect, select the first equation in the solution process that is not true, Osuming that the prior equation is true. ► x + 4 = 2x + 11 3x + 4 = 11 3x = 15 x = 5 The solution is correct.
Answer: Yes the soultuion is corrrect
Step-by-step explanation:
This is the same information from the prior question.
Skidding distance can be determined using the formula d space equals space 2 square root of 5 x end root where d is the distance (in feet) and x is the speed (in miles per hour).
Calculate the approximate speed of the car when the skid marks are 20 feet long.
The approximate speed of the car when the skid marks are 20 feet long is 8.944 miles per hour.
What is importance of a formula?A formula is an expression in mathematics that denotes a connection or rule. In mathematics, science, and engineering, numbers and variables are described and computed using formulas. They can be stated verbally or symbolically, such as in verbal descriptions, or mathematically or symbolically, such in algebraic equations. Formulas can be applied to evaluate data, solve issues, and make predictions. They are utilised in a variety of disciplines, including physics, chemistry, biology, finance, and statistics, and are crucial for comprehending and explaining the physical world.
The skidding distance is given by the formula:
d = 2 * √(5x)
Isolating the value of x we have:
x = (d / (2 * √(5)))
Substituting the value of d = 20 we have:
x = (20 / (2 * √(5))) ≈ 8.944 miles per hour
Hence, the approximate speed of the car when the skid marks are 20 feet long is 8.944 miles per hour.
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PLEASE HELP!! SPECIAL RIGHT TRIANGLES I"LL GIVE YOU BRAINLIEST OR WHATEVER
Using the trigonometry function, the value of y = 3√2.
What exactly does math trigonometry mean?The area οf mathematics knοwn as trigοnοmetry is cοncerned with analyzing the relatiοnships between the sides and angles οf a right-angled triangle. A trigοnοmetric ratiο called the sides' ratiο serves as a representatiοn οf the relatiοnship. Trigοnοmetric ratiοs cοme in six varieties: sine, cοsine, tangent, cοtangent, secant, and cοsecant.
Trigοnοmetric functiοns are used tο calculate the lengths and angles οf geοmetric fοrms frοm knοwn οr measured angles. As a result οf the need tο cοmpute angles and distances in numerοus fields, including astrοnοmy, mapmaking, surveying, and artillery range finding, trigοnοmetry was develοped.
Since it is a 90-60-30 triangle, y is s √2
s = 3
so, y = 3√2
y = 3√2
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solve the system of equations -3x-5y=14 and 7x+7y=0 by combining the equations.
Answer:
x=7 & y=-7
Step-by-step explanation:
from the 2nd equation 7x+7y=0 we can get that
x=-y
substitute that into the first equation to get
3y-5y=14
-2y=14
y=-7
then
x=7
The product of two positive number is 108. If one one number is treble of other number, find those number
A city has 5 new houses for every 8 old houses. If there are 30 new houses in the city, how many old houses are there?
Answer: 48 old houses
Step-by-step explanation:
Ratio 5:8
Ratio 30:?
30/5=6
8 x 6= ...
48!!!
30:48
did this make sense??
If $4000 is invested at 7% interest per year compounded continuously, how long will it take to double the original investment?
it will take about 9.90 years to double the initial investment of $4000 at 7% continuous compound interest.
We can use the formula for continuous compound interest:
[tex]A = Pe^(rt)[/tex]
where A is the amount of money after time t, P is the principal (initial investment), r is the interest rate, and e is the mathematical constant approximately equal to 2.71828.
To find the time it takes to double the initial investment, we want to solve for t when A = 2P (twice the initial investment). Substituting the given values, we have:
[tex]2P = Pe^(rt)[/tex]
Dividing both sides by P, we get:
[tex]2 = e^(rt)[/tex]
Taking the natural logarithm (ln) of both sides, we have:
[tex]ln(2) = ln(e^(rt))[/tex]
ln(2) = rt * ln(e)
Since ln(e) = 1, we can simplify further to:
ln(2) = rt
Solving for t, we get:
t = ln(2) / r
Substituting the given values, we have:
t = ln(2) / 0.07
t ≈ 9.90 years
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As given on the STEPS Engineers Ireland website ( Engineers Week, Secondary School section) how many free engineering activities are available to download by teachers for classroom use?
Which two points would a line of fit go through to best fit the data? (6, 4) and (9, 1) (3, 5) and (10, 1) (1, 8) and (10, 1) (1, 5) and (7, 3)
The two points that a line of fit would go through to best fit the data are (3, 5) and (10, 1).
We need to find which pair of points have a line passing through them that best fits the data. One way to do this is to calculate the slope of the line passing through each pair of points and choose the pair of points with the slope closest to the average slope.
The slope of a line passing through two points [tex](x_1, y_1)[/tex] [tex](x_2, y_2)[/tex] is given by the formula:
slope = [tex]\frac{(y_2 - y_1)}{ (x_2 - x_1)}[/tex]
Using this formula, we can calculate the slopes of the lines passing through each pair of points:
Pair 1: slope = [tex]\frac{(1 - 4)}{(9-6)}[/tex] = -1
Pair 2: slope = [tex]\frac{(1 - 5)}{(10-3)}[/tex] = -0.67
Pair 3: slope = [tex]\frac{(1 - 8) }{(10-1)}[/tex] = -0.88
Pair 4: slope =[tex]\frac{ (3 - 5)}{(7-1)}[/tex] = -0.33
The average slope is:
average slope = [tex]\frac{(-1 + (-0.67) + (-0.88) + (-0.33))}{4}[/tex] = -0.72
The pair of points with the slope closest to the average slope is pair 2, with a slope of -0.67. Therefore, the two points that a line of fit would go through to best fit the data are (3, 5) and (10, 1).
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Your friend estimates that a bookcase is 2 1/2 feet wide. The actual width is 2/3 foot longer. What is the width of the bookcase?
1 Divide the given polynomial by the given monomial. (1) (5x² - 6x) ÷ 3x C C 31 2-2-2 (V)
The solution to the expression (5x² - 6x) / 3x is (5/3)x - 2
What is an equation?An equation is an expression that shows the relationship between numbers and variables using mathematical operators such as addition, subtraction, exponent, multiplication and division.
Equations can be classified based on degree as linear, quadratic, cubic and so on.
Given the polynomial:
5x² - 6x
Dividing the polynomial by the monomial 3x:
= (5x² - 6x) / 3x
= [x(5x - 6)] / 3x
= (5x - 6) / 3
= (5/3)x - 2
The solution to the equation is (5/3)x - 2
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In an obstacle course, participants climb to the top of a tower and use a zip line to travel across a mud pit. The zip line extends from the top of a tower to a point on the ground 48. 2 feet away from the base of the tower. The angle of elevation of the zip line is 33°. Estimate the length of the zip line to the nearest tenth of a foot
The estimated length of the zip line is 62.8 feet.
The length of a zip line can be estimated using the formula for the length of a side of a right triangle, which is the square root of the sum of the squares of the two shorter sides. In the case of the obstacle course, the two shorter sides are the height of the tower and the horizontal distance from the base of the tower to the point on the ground.
To calculate the length of the zip line, we first need to calculate the height of the tower. The angle of elevation of the zip line is 33°, so we can use trigonometry and the Law of Sines to calculate the height of the tower. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is the same for all sides of the triangle. Since we know the angle of elevation (33°) and the horizontal distance (48.2 feet), we can calculate the height of the tower using the formula:
h = (48.2 * sin(33°)) / sin(90°)
h = 42.7 feet
Now we can calculate the length of the zip line using the formula for the length of a side of a right triangle:
[tex]L = sqrt(h^2 + d^2)L = sqrt(42.7^2 + 48.2^2)L = 62.8 feet[/tex]
Rounding to the nearest tenth of a foot, the length of the zip line is 62.8 feet.
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2. If twice a number is decreased by 13, the result is 9. What is the number?
Answer:
Let's start by translating the sentence into an equation using algebra.
Let's say the number is represented by "x".
According to the problem:
"Twice a number" can be written as "2x".
"If twice a number is decreased by 13" can be written as "2x - 13".
"The result is 9" can be written as "= 9".
Putting it all together, we get the equation:
2x - 13 = 9
Now we can solve for x:
2x = 22
x = 11
Therefore, the number is 11.
Pleas can someone help!
Answer:
99.6 cm²
Step-by-step explanation:
You want to know the area of a sector that has radius 6 cm, and subtends an arc of 317°.
Sector areaThe formula for the area of a sector is ...
A = 1/2r²θ . . . . . . where θ is the central angle in radians
ApplicationA = 1/2(6 cm)²(317·π/180) ≈ 99.6 cm²
The area of the sector is about 99.6 square centimeters.
Find the slope show your work you don't have to do them all I just need some of them done please
Answer: 1. Slope equals 3/4 or 0.75, 2. Slope equals -5/2 or -2.5
Step-by-step explanation:
1. Subtract y coordinates 4 and -2 to get 6. Then subtract x coordinates 4 and -4 to get 8, so that is equal to 6/8. Then simplify to get 3/4 or 0.75
2. Subtract y coordinates 10 and -5 to get -15. Then subtract x coordinates 0 and -6 to get 6, so that is equal to-15/6. Then simplify to get -5/2 or -2.5.
The pressure that a box exerts on a shelf is 200 N/m². The force that the box exerts on the shelf is 180 N. Work out the area of the base of the box. If your answer is a decimal, give it to 1 d.p. Question attached!
Step-by-step explanation:
To solve this problem, we can use the formula for pressure:
pressure = force / area
We are given the pressure and the force, so we can rearrange the formula to solve for the area:
area = force / pressure
Substituting the values we have:
area = 180 N / 200 N/m²
area = 0.9 m² (to 1 decimal place)
Therefore, the area of the base of the box is 0.9 square meters.
What is the surface area of the three-dimensional figure represented by this net?
The net pattern has a square base of side lengths 3 inches, 3 inches, 3 inches and 3 inches are attached with four congruent triangles of height 8 inches.
Use the on-screen keyboard to type the correct number of square inches in the box below.
$$
The surface area of the three-dimensional figure represented by this net is 54 square inches, which is calculated by multiplying the area of the base (9 square inches) by the height of the four congruent triangles (8 inches).
The surface area of the three-dimensional figure represented by this net is 54 square inches. To calculate this, first we need to find the area of the base. The base is a square, with side lengths of 3 inches each. Therefore, the area of the base is 3 x 3 = 9 square inches. Next, we need to find the height of the four congruent triangles. The height is 8 inches. Finally, we can calculate the surface area by multiplying the area of the base (9 square inches) by the height of the four congruent triangles (8 inches). This gives us a total surface area of 54 square inches.
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f(x)=-2x^2+6x+1 what is the rate of change
The rate of change of the function f(x) = -2x² + 6x + 1 at x = 1 is approximately 0.302.
What is the definition of a function?
In mathematics, a function is a rule that assigns to each element in a set (called the domain) exactly one element in another set (called the range). In other words, a function takes an input value and produces a unique output value. The input value is usually represented by the variable x, while the output value is represented by the variable y or f(x).
Now,
The rate of change of a function is the slope of the line that connects any two points on the graph of the function. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
slope = (y2 - y1)/(x2 - x1)
So to find the rate of change of the function f(x) = -2x² + 6x + 1, we need to choose two values of x and find the corresponding values of f(x), and then calculate the slope of the line passing through these two points.
Let's choose two values of x that are close to each other, say x1 = 1 and x2 = 1.01. Then, we have:
f(x1) = -2(1)² + 6(1) + 1 = 5
f(x2) = -2(1.01)² + 6(1.01) + 1 = 5.0302
Now, we can calculate the slope of the line passing through these two points:
slope = (f(x2) - f(x1))/(x2 - x1) = (5.0302 - 5)/(1.01 - 1) = 0.302
So,
the rate of change of the function f(x) = -2x² + 6x + 1 at x = 1 is approximately 0.302. Note that the rate of change of a function is not constant, but varies with the value of x.
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4) Two cars leave the same parking lot, with one heading north and the other heading east.
After several minutes, the northbound car has traveled 8 kilometers, and the eastbound car
has traveled 2 kilometers. Measured in a straight line, how far apart are the two cars? If
necessary, round to the nearest tenth.
Video
Answer:
8.2km
Step-by-step explanation:
North and East are perpendicular directions;
If you illustrate the distances covered in each direction and join the two lines, it will form a right-angle triangle;
This means pythagoras applies:
(8)² + (2)² = 64 + 4 = 68
The distance apart is, therefore:
[tex] \sqrt{68} = 8.2462112512 [/tex]
27. If T and P are positive numbers,which of the following is always false?
A. P-T>O
B. P+T=0
C. P + T = 2P
D. P+T> P
Answer:
B
Step-by-step explanation:
The only way for two numbers to = zero when they're added together is if one is negative. There are no two positive numbers that when added together will equal 0, therefore if p and t are positive, their sum cannot be 0
Fifty pairs of individuals playing in a bridge tournament have been seeded 1,. , 50. In the first part of the tournament, the 50 are randomly divided into 25 east-west pairs and 25 north-south pairs.
(a) What is the probability that x of the top 25 pairs end up playing east-west?
(b) What is the probability that all of the top five pairs end up playing the same direction?
(c) If there are 2n pairs, what is the pmf of X = the number among the top n pairs who end up playing east-west? What are E(X) and V(X)?
a .The probability that x of the top 25 pairs play east-west is: P(X = x) = (25 choose x) / (50 choose 25) , b. The probability that the top 5 pairs all play in the same direction is: P(all 5 play the same direction) = (1/2) * (24/49) * (23/48) * (22/47) * (21/46), c. The pmf of X is given by P(X = x) = [(n choose x) * (n choose n - x)] / [(2n choose n)], and the expected value and variance of X are E(X) = n/2 and V(X) = (n² - 1)/12.
P(X = x) = [(n choose x) * (n choose n - x)] / [(2n choose n)] yields the pmf of X. , and X's expected value and variance are, respectively, E(X) = n/2 and V(X) = (n² - 1)/12.
(a) Let X be the number of top 25 pairs that play east-west. Since there are 25 east-west pairs and 25 north-south pairs, the total number of ways to choose 25 pairs to play east-west is (50 choose 25) = 50! / (25! * 25!). Similarly, the total number of ways to choose x top pairs to play east-west is (25 choose x) * (25 choose 25 - x) = (25 choose x), since the remaining 25 pairs automatically play east-west. Therefore, the probability that x of the top 25 pairs play east-west is:
P(X = x) = (25 choose x) / (50 choose 25)
(b) The probability that the top pair plays east-west is 25/50 = 1/2. Once the top pair has been assigned a direction, the probability that the second pair plays in the same direction is 24/49, the probability that the third pair plays in the same direction is 23/48, and so on. Therefore, the probability that the top 5 pairs all play in the same direction is:
P(all 5 play the same direction) = (1/2) * (24/49) * (23/48) * (22/47) * (21/46)
(c) Let X be the number of top n pairs that play east-west, where there are 2n pairs total. Then X can take on values from 0 to n. To find the pmf of X, we can use the same method as in part (a):
P(X = x) = [(2n - n) choose x] / [(2n) choose n]
= [(n choose x) * (n choose n - x)] / [(2n choose n)]
The expected value of X is:
E(X) = sum[x * P(X = x), {x, 0, n}]
= sum[x * (n choose x) * (n choose n - x) / (2n choose n), {x, 0, n}]
Using the identity sum[x * (n choose x) * (n choose k - x), {x, 0, k}] = k * (n choose k), we can simplify this to:
E(X) = n/2
The variance of X is:
V(X) = E(X²) - E(X)²
= sum[x² * P(X = x), {x, 0, n}] - (n/2)²
= sum[x * (n choose x) * (n choose n - x) / (2n choose n), {x, 0, n}] - (n/2)²
= (n² - 1)/12
Therefore, the pmf of X is given by P(X = x) = [(n choose x) * (n choose n - x)] / [(2n choose n)], and the expected value and variance of X are E(X) = n/2 and V(X) = (n² - 1)/12, respectively.
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I need the answers quick please !!
The number of triangles that can be formed from a common vertex on the polygon shown is three triangles.
How to find the number of triangles ?To form triangles on polygons, you can draw diagonals connecting the vertices of the polygon. A diagonal is a line segment that connects two non-adjacent vertices of a polygon. By drawing diagonals, you can create triangles within the polygon.
From a hexagon, we can draw three diagonals from each vertex. So, from a common vertex, we can draw three diagonals and form three triangles. Since there are six vertices in a hexagon, we can choose any one of these vertices as the common vertex and form three triangles.
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What is the equation of the line?
Answer:
[tex]y = -\dfrac{1}{4}x + 1[/tex]
Step-by-step explanation:
Slope of the line can be found by taking two points on the line, (x1, y1) and (x2, y2)
Find the difference in y coordinates, y2 - y1 (called rise)
Find the difference in corresponding x coordinates x2 - x1 (called run)
rise/run gives the slope
Two distinguishable points on the line graph are (0, 1) and (4, 0)
rise = 0 - 1 = - 1
run = 4 - 0 = 4
Slope = -1/4
The equation of the line in general form is
y = mx + b
where
m = slope
b = y-intercept (value of y when x = 0)
Here m = -1/4
b = 1
So equation of the line is
[tex]y = -\dfrac{1}{4}x + 1[/tex]
rick has 3x and helen has twice that amount they save 3 each per week what amount will be the sum of their money 4 weeks from now
Hence the amount of savings of Rick and Halens is = 9x+24.
When we write any linguistic statement in mathematics form called mathematical expression.
Rick has 3x to start with
Let w = the number of weeks he will save.
Savings = 3x + 3*w
Let w = 4
Savings = 3x+3*4
Savings = 3x + 12
Halen has twice Rick's amount,
Let w = the number of weeks she will save.
Savings=6x+3w
let w=4
savings=6x+12
Now we come to the question. It seems to want the total combining both of them.
Rick + Halen = 3x+12+6x+12 = 9x+24
Hence the amount of savings of rick and Halens is = 9x+24.
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BRAINLIESTT ASAP! PLEASE HELP ME :)
Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.
(1) If you use a pencil you can erase mistakes.
(2) If you can erase mistakes your paper will be neater.
(3) If you use a pencil your paper will be neater.
A. Invalid
B. Yes; Law of Detachment
C. Yes; Law of Syllogism
Statement (3) follows from statements (1) and (2) by the Law of Syllogism. So, the correct option c) which is Yes; Law of Syllogism
Here's the reasoning:
(1) If you use a pencil you can erase mistakes.
(2) If you can erase mistakes your paper will be neater.
Therefore, by the Law of Syllogism,
(3) If you use a pencil your paper will be neater.
The Law of Syllogism states that if we have two conditional statements (if A then B) and (if B then C), we can logically conclude a third conditional statement (if A then C). That's exactly what we've done here since statement (1) is the first conditional statement, statement (2) is the second conditional statement, and statement (3) is the conclusion that follows from applying the Law of Syllogism to the first two statements.
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1. (20 marks) A survey database shows that 20 percent of the adults in country A have been tested for HIV at some point in their life. (a) Suppose that an adult who has been tested for HIV at some point in his life is needed in a study. Some adults are randomly selected from the population of country A in the selection process until the one who has been tested for HIV is found. Let X be the number of adults selected in this process. Show that the cumulative probability P(X≤x)=1−0.8 x . (4 marks) (b) Using the formula of P(X≤x) in (a), find the probability that the number of adults selected in the process are: (i) Three, (ii) Less than five, (iii) Between five and nine, inclusive, (iv) More than five, but less than 10, (v) Six or more. (5 marks) (c) Find the mean and variance of the number of people selected in the above process. (3 marks) (d) Consider a simple random sample of 18 adults. Find the probability that the number of adults who have been tested for HIV in the sample would be: (i) Three, (ii) Less than five, (iii) Between five and nine, inclusive, (iv) More than five, but less than 10 , (v) Six or more. ( 5 marks) (e) Find the mean and variance of the number of people tested for HIV in the sample. (3 marks)
From the given data provided, the probability that number of adults selected in process are 3, less than 5, between 5 and 9, more than 5 but less than 10, 6 or more is 0.096, 0.41, 0.336, 0.232, 0.168 and mean of number of people tested for HIV is 3.6.
(a) The probability that an adult has not been tested for HIV is 1-0.20=0.80. Therefore, the probability that the first adult selected has not been tested for HIV is 0.80. The probability that the first and second adults selected have not been tested for HIV is 0.80 x 0.80=0.64. Continuing in this way, we see that the probability that X adults must be selected until one who has been tested for HIV is found is given by:
P(X≤x) = 1 - 0.80ˣ
(b) Using the formula from (a), we can find the probabilities as follows:
(i) P(X=3) = P(X≤3) - P(X≤2) = (1-0.80³) - (1-0.80²) = 0.096
(ii) P(X<5) = P(X≤4) = 1-0.80⁴ = 0.41
(iii) P(5≤X≤9) = P(X≤9) - P(X≤4) = (1-0.80⁹) - (1-0.80⁴) = 0.336
(iv) P(6≤X<10) = P(X≤9) - P(X≤5) = (1-0.80⁹) - (1-0.80⁵) = 0.232
(v) P(X≥6) = 1 - P(X≤5) = 1 - (1-0.80⁵) = 0.168
(c) To find the mean and variance of X, we first note that X follows a geometric distribution with parameter p=0.20. Therefore, the mean and variance of X are:
Mean of X = E(X) = 1/p = 1/0.20 = 5
Variance of X = Var(X) = (1-p)/p² = 0.8/0.04 = 20
(d) To find the probabilities for a random sample of 18 adults, we use the binomial distribution with parameters n=18 and p=0.20. The probabilities are:
(i) P(X=3) = (18 choose 3) x 0.20³ x 0.80¹⁵ = 0.236
(ii) P(X<5) = P(X≤4) = Σ(18 choose x) x 0.20ˣ x 0.80⁽¹⁸⁻ˣ⁾ for x=0 to 4 = 0.678
(iii) P(5≤X≤9) = Σ(18 choose x) x 0.20ˣ x 0.80⁽¹⁸⁻ˣ⁾ for x=5 to 9 = 0.286
(iv) P(6≤X<10) = Σ(18 choose x) x 0.20ˣ x 0.80⁽¹⁸⁻ˣ⁾ for x=6 to 9 = 0.207
(v) P(X≥6) = 1 - P(X≤5) = 1 - Σ(18 choose x) x 0.20ˣ x 0.80⁽¹⁸⁻ˣ⁾ for x=0 to 5 = 0.322
(e) The mean and variance of X for a sample of 18 adults are:
Mean of X = E(X) = n x p = 18 x 0.20 = 3.6
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Find the value of x in the regular octagon when each angle is represented by the expression (8x+23)°
The value of x in the regular octagon is 14°. The solution has been obtained by using arithmetic operations.
What are arithmetic operations?
All real numbers are claimed to be able to be described by the four basic operations, sometimes known as "arithmetic operations". Quotient, product, sum, and difference are the next four mathematical operations after division, multiplication, addition, and subtraction.
We are given that the measure of each angle is (8x + 23)°.
We know that the sum of angles of an octagon is 1080°.
So, using the arithmetic operation, we get
⇒ 8 * (8x + 23)° = 1080°
⇒ 64x + 184 = 1080°
⇒ 64x = 896°
⇒ x = 14°
Hence, the value of x in the regular octagon is 14°.
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