Using complementary angles, the value of x = 10.3
What are complementary angles?Complementary angles are angles in which the sum add up to 90°.
How to find the value of x in the given right triangle?Since in a right triangle, sin(8x - 9) = cos(x + 6). To solve the problem, we proceed as follows.
sin(8x - 9) = cos(x + 6).
Given that cosФ = sin(90 - Ф), we have that
cos(x + 6) = sin[90 - (x + 6)]
So, susbstituting this into the equation, we have that
sin(8x - 9) = cos(x + 6)
sin(8x - 9) = sin[90 - (x + 6)]
This implies that
8x - 9 = 90 - (x + 6)
8x - 9 = 90 - x - 6
8x - 9 = 84 - x
Adding x to both sides of the equation, we have
8x + x - 9 = 84 - x + x
9x - 9 = 84 + 0
9x - 9 = 84
Adding 9 to both sides of the equation, we have that
9x - 9 + 9 = 84 + 9
9x + 0 = 93
9x = 93
x = 93/9
x = 10.33
x ≅ 10.3
So, the value of x = 10.3
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(v) (5-6i)+(3+4i) - (5-7)
Answer:
Step-by-step explanation
480,000 in scientific notation.
Answer:4.8×105 g
Step-by-step explanation:
a student has only a few hours to prepare for two different exams tomorrow morning. the above table shows alternative possible exam outcomes with three alternative uses of the student's time. the opportunity cost of scoring an 84 on the history exam rather than 76 is:
The opportunity cost of scoring a 84 on the economics exam rather than a 76 is 15 points on the History exam.
How to determine the opportunity cost?Opportunity Cost is what a person sacrifices when they choose one option over another. It is calculating by subtracting the return of the best forgone option with the return of the chosen option. The outcome could be beneficial or prejudicial, depending on the case.
In the example (refer to the attached table), if the student chooses to score 94 in the economics exam then the student will get 76 in the History test. Thus, The opportunity cost of getting 94 instead of 77 in the Economics test, implies getting 76 instead of 91 in the History exam. It implies:
Opportunity cost of the History exam = 91-76
Opportunity cost of the History exam = 15
Therefore the opportunity cost of scoring 94 on the Economics exam rather than a 77 is 15 points on the History test.
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What is the length of the apothem of a regular hexagon with side length of 8 ft?
The length of the apothem of a regular hexagon is 4√3
How to determine the length of the apothemThe apothem of a regular hexagon is the distance from the center of the hexagon to the midpoint of one of its sides.
To find the length of the apothem, we can use the formula:
apothem = (side length) / (2 * tan(180/n))
where n is the number of sides of the polygon.
In this case, n = 6 (since the hexagon is regular).
So, to find the apothem of a regular hexagon with a side length of 8 ft, we can plug these values into the formula:
apothem = (8 ft) / (2 * tan(180/6))
This gives
apothem = 8 ft / (2 * tan(30))
Evaluate
apothem = 4√3 ft
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In this diagram, circle A has radius = 11.5 and BC = 13.1. BC is a tangent line, calculate the distance of CD.
The distance of CD is 5.93.
What is Circle?Circle is a two dimensional figure which consist of set of all the points which are at equal distance from a point which is fixed called the center of the circle.
Given a circle.
Radius of circle = 11.5
AB = AD = 11.5
Since the tangent line is perpendicular to the radius of the circle at the point of tangency, ABC is a right angled triangle.
AC² = AB² + BC²
= 11.5² + 13.1²
= 303.86
AC = 17.432
CD = AC - AD
= 17.432 - 11.5
= 5.93
Hence the distance of CD is 5.93.
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create the following matrices with the help of the matrix generation functions eye , ones, diag and triu. see the on-line help on these functions if required (i.e. help eye). do not enter the matrices explicitly. each matrix should be generated with a single command. m
The required solution is above with the help of matrix generation.
Now, According to the question:
A statement is a declarative sentence that is either true or false but not both. A statement is sometimes called a proposition. The key is that there must be no ambiguity. To be a statement, a sentence must be true or false, and it cannot be both.
With the help of the matrix generation functions eye , ones, diagonal and trigonal .
Command to generate matrix such as M,N,P,Q
M=6*eye(3)
N= triu (5*ones(3))
P=7*ones (3,2)
Q=diag ([2 3 4])
Therefore, the required solution is above with the help of matrix generation.
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The given question is incomplete, complete question is:
Create matrices with eye, ones, diag and triu Create the following matrices with the help of the matrix generation functions eye , ones, diag and triu. See the on-line help on these functions if required (i.e. help eye)M =[6 0 00 6 00 0 6]N =[5 5 50 5 50 0 5 ]P =[7 77 77 7]Q =[2 0 00 3 00 0 4]
The following inequalities form a system. y is greater than or equal to two-thirds times x plus 1 y is less than negative one-fourth times x plus 2 Which ordered pair is included in the solution to this system? (−6, 3.5) (−6, −3) (−4, 3) (−4, 4)
Answer: The inequalities are y ≥ (2/3)x + 1 and y < (-1/4)x + 2 and (−6, −3) satisfied the inequality.
What exactly is inequality?
A variation between two values indicates whether one is smaller, larger, or otherwise unrelated to the other.
In other words, inequality is the inverse of equality. For example, if 2 = 2, it is equal; however, if I say 3 = 6, it is incorrect; the correct expression is 3 6.
Step-by-step explanation:As per the given phrase,
y is greater than or equal to two-thirds times x plus 1 ⇒ y ≥ (2/3)x + 1
y is less than negative one-fourth times x plus 2 ⇒ y < (-1/4)x + 2
Substitute, point (−6, −3)
-3 ≥ (2/3)(-6) + 1
-3 ≥ -3
Substitute in second inequality,
-3 < (-1/4)(-6) + 2
-3 < 7/2
B = 1 4 1 2
0 1 3 -4
0 2 6 7
2 9 5 -1
Can every vector in R^4 be written as a linear combination of the columns of the matrix B above? Do the columns of B span R^3
Yes, every vector in R^4 can be written as a linear combination of the columns of the matrix B.
This can be proven using the definition of linear combinations, which is as follows: a vector v in R^n can be written as a linear combination of vectors v1, v2, ..., vk if and only if there exist scalars c1, c2, ..., ck such that v = c1*v1 + c2*v2 + ... + ck*vk. With this definition, we can prove that the columns of B span R^4 by showing that for any vector v in R^4, there exist scalars c1, c2, c3, c4 such that v = c1*B1 + c2*B2 + c3*B3 + c4*B4, where B1, B2, B3, B4 are the columns of the matrix B above. To prove this, we can simply solve the system of equations given by c1*B1 + c2*B2 + c3*B3 + c4*B4 = v for c1, c2, c3, c4. This can be done by using Gaussian elimination, which will yield a set of scalars c1, c2, c3, c4 that can be used to express any vector in R^4 as a linear combination of the columns of the matrix B, thus proving that the columns of B span R^4.
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a student sets up the following equation to convert a measurement. (the ? stands for a number the student is going to calculate.) fill in the missing part of this equation. (39 . N/cm) . ____ = ? N/mm
The resulting answer is 39000 N/mm, this equation can be used to convert a given measurement from N/cm to N/mm.
39000/1 = ? N/mm
Multiply 39 by 1000 to convert centimeters to millimeters.
39000/1 = ? N/mm
Divide the measurement 39000 N/cm by 1 to convert it to N/mm.
The answer is 39000 N/mm.
To convert a measurement from N/cm to N/mm, the equation that needs to be set up is 39000/1 = ? N/mm. This equation can be broken down into two steps. First, the measurement needs to be multiplied by 1000 to convert centimeters to millimeters. Then, 39000/1 needs to be divided to convert the measurement from N/cm to N/mm. The resulting answer is 39000 N/mm. Therefore, this equation can be used to convert a given measurement from N/cm to N/mm.
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I don't understand how to calculate IQR here. Is 2 the median?
A statement which is false about the distribution of Kobe’s streak lengths from the 2009 NBA finals include the following: E. The shortest streak is of length 1.
How to determine the false statement?In this scenario, the distribution of Kobe’s streak lengths from the 2009 NBA finals is plotted on a histogram with the length represented by the x-coordinate (x-axis) while the count is represented by the y-coordinate (y-axis).
By critically observing the histogram, we can reasonably infer and logically deduce the following true statements:
The distribution of Kobe’s streak lengths is right skewed and unimodal because it has only one clear peak (most frequent value).Since the median of the distribution of Kobe’s streak lengths is at 0, the typical length of a streak is equal to 0.The interquartile range (IQR) of the distribution is 1 i.e IQR = Q₃ - Q₁.In conclusion, the shortest streak lengths is 0 while the longest streak of baskets is of length 4.
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Complete Question:
Which of the following is false about the distribution of Kobe’s streak lengths from the 2009 NBA finals.
A. The distribution of Kobe’s streaks is unimodal and right skewed.
B. The typical length of a streak is 0 since the median of the distribution is at 0.
C. The IQR of the distribution is 1.
D. The longest streak of baskets is of length 4.
E. The shortest streak is of length 1
PLEASE HELP!!!!! Complete the two-column proof.
The equation will be 5x + 7x = 180. Then the value of the variable 'x' will be 15.
What is an angle?The inclination is the separation seen between planes or vectors that meet. Degrees are another way to indicate the slope. For a full rotation, the angle is 360°.
Supplementary angle - Two angles are said to be supplementary angles if their sum is 180 degrees.
∠TSR = 5x and ∠UVW = 7x
Angle ∠TSR and angle ∠UVW are the supplementary angles. Then the equation is given as,
5x + 7x = 180
12x = 180
x = 180 / 12
x = 15
The equation will be 5x + 7x = 180. Then the value of the variable 'x' will be 15.
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Penny created a circle graph to show what percentage of students in her class walk to school, ride the bus, or ride in a car.
What percent of students in Penny’s class ride the bus to school?
HELPPPP ASAPPPPP
Answer:
45
Step-by-step explanation:
100-25-30=45
A sample was taken of the salaries of four employees from a large company. The following are their salaries (in thousands of dollars) for this year. 33 31 24 36 The variance of their salaries is A. 5.1. B. 31. C. 26.
The variance of the four employees' salaries is 5.1, meaning that the difference in their salaries is 5.1 thousand dollars.
Step 1: Find the mean of the four salaries
33 + 31 + 24 + 36 = 124
124 ÷ 4 = 31
Mean = 31
Step 2: Find the difference between each salary and the mean
33 - 31 = 2
31 - 31 = 0
24 - 31 = -7
36 - 31 = 5
Step 3: Square each of the differences
2² = 4
0² = 0
(-7)² = 49
5² = 25
Step 4: Add up the squared differences
4 + 0 + 49 + 25 = 78
Step 5: Divide the sum by the number of salaries (minus 1)
78 ÷ (4 - 1) = 78 ÷ 3 = 26
Step 6: Take the square root of the variance
√26 = 5.1
Variance = 5.1
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consider the following argument, premise 1: if the sum of the digits in (student id) is divisible by 3 then (student id) is divisible by 3. premise 2: the sum of the digits in (student id) is divisible by 3. conclusion: (student id) is divisible by 3. substitute your eight-digit numeric student id for (student id) in the above argument and determine whether the argument, filled-in with your student id, is sound or unsound. justify your answer and include your student id in your explanation.
The argument, filled-in with the student ID 12345678, is sound because the two premises are true and the conclusion follows logically from them. The sum of the digits in 12345678 is 36, which is divisible by 3, so the first premise is true.
My student ID is 12345678.
The argument, filled-in with my student ID, is sound. This is because the two premises are true and the conclusion follows logically from them. The sum of the digits in 12345678 is 36, which is divisible by 3, so the first premise is true. The second premise is also true since it states that the sum of the digits in 12345678 is divisible by 3, which is the case. Therefore, the conclusion that 12345678 is divisible by 3 is logically valid from the two premises.
Student ID: 12345678
The argument, filled-in with the student ID 12345678, is sound because the two premises are true and the conclusion follows logically from them.
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The perimeter of a certain rectangle is 26 m. Also, the difference between the length of the rectangle and the breath is 3 m. Find the length and the breadth of the rectangle.
A rectangle having perimeter 26 m has a length of 8m and breadth of 5m.
What is perimeter?
The complete length of a shape's boundary is referred to as the perimeter in geometry. A shape's perimeter is calculated by adding the lengths of all of its sides and edges. Its dimensions are expressed in linear units like centimetres, metres, inches, and feet.
The value for perimeter of a rectangle is - 26 m
Let x be the length of the rectangle and y be the breadth.
The formula for perimeter of rectangle is 2(l + b)
From the given information, it is known that -
Perimeter = 2(x + y) = 26 m...(1)
Difference between the length and breadth = x - y = 3 m....(2)
Use the first equation to solve for one of the variables in terms of the other -
2x + 2y = 26
x + y = 13....(3)
Add equation (2) and (3) -
x + y + x - y = 13 + 3
2x = 16
x = 16/2
x = 8 m
Now substitute the value of x in equation (2) -
x - y = 3
8 - y = 3
y = 8 - 3
y = 5 m
Therefore, the length is 8 m and the breadth is 5 m.
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Triangle AA'B'C" is the result of dilating AABC about point P by a scale factor of 2.
Answer:
FalseFalse==============================================
Explanation for problem 1
The center of dilation is point P, which is not located on line AB. This means line A'B' will move further away from P after applying the scale factor. It means segments AB and A'B' are not on the same line.
Side note: Side BC is through line P, so segment B'C' shares the same line with BC.
---------------------------
Explanation for problem 2
The scale factor enlarges the side lengths. In this case, the scale factor 2 will double each side. This in turn doubles the perimeter.
perimeter of A'B'C' = 2*(perimeter of ABC)
new perimeter = 2*(old perimeter)
Therefore, the perimeters are not the same.
The average pencil has 5,425 pencil shavings. How many pencils are found in 76,450 pencil shavings?
Answer:
14
Step-by-step explanation:
We can divide 76,450 shavings by the average shavings per pencil, 5,425 to get an answer of 14.09... This can be rounded to 14.
What is the total area, in square inches, of the shaded sections of the trapezoid below? 5 in 6.3 in 8 in 5.4 in
The total area of the shaded sections of the trapezoid in square inches is 32.76 square inches.
What is area?An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely fill the entire surface of an enclosed figure. The square unit, which is frequently expressed as square inches, square feet, etc., is the accepted unit of area.
The shaded sections in the trapezoid are triangles.
The area of the triangle is given by the following formula:
A = 1/2 (b)(h)
For the first triangle, b = 5 and h = 6.3. Substituting the value in the formula we have:
A1 = 1/2 (5)(6.3)
A1 = 15.75
For the second triangle, b = 5.4 and h = 6.3. Substituting the value in the formula we have:
A2 = 1/2 (5.4)(6.3)
A2 = 17.01
The total area of the shaded region of the trapezoid is calculated by adding the area of the individual shaded portion:
A = A1 + A2
A = 15.75 + 17.01
A = 32.76
Hence, the total area of the shaded sections of the trapezoid in square inches is 32.76 square inches.
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EACH COPAYMENT IS 20% OF THE COST FIND THE COPAYMENT AND BENIFIT AMOUNT 4,200,00+0=3,072.00
The copayment amount, given that it is 20 % of the amount is $ 840
The benefit amount is $ 3, 360
How to find the copayment amount ?A copayment for health insurance is a defined dollar amount that is determined by the insurance plan and used to divide the cost of covered treatments between the plan and the consumer.
The copayment amount is 20 % of the cost which is $ 4, 200 so the copayment is:
= 20 % x 4, 200
= $ 840
This then means that the benefit amount would be;
= Cost - copayment
= 4, 200 - 840
= $ 3, 360
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In order to produce my movie, I need to shoot a scene for a high-speed chase and I need to shoot a scene for a bank robbery. The film production crew will be putting the movie together. I have the stunt crew for the chase scene from 10 am to 11 am and I have the stunt crew for the robbery from 6pm to 7 pm. Will the order I shoot the scenes in matter? What mathematical property did you use to answer this question?
The order in which you shoot the scenes will not matter in terms of the final outcome of the movie.
The mathematical property that one can use to answer this question is scheduling.
How to illustrate the information?From the information, in order to produce the movie, one needs to shoot a scene for a high-speed chase and need to shoot a scene for a bank robbery.
Here, the film production crew will be putting the movie together. However, it may impact the logistics of the production process and the scheduling of the crews. It is not a mathematical property that I used to answer this question. It is simply a matter of logistics and scheduling.
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mation.
2) (x-3)(x-8)= 0
Answer:
x-3 =0 or x-8 =0
x= +3 or x = +8
Which figures have rotational symmetry?
evaluate x^3 - 2x^2 +3x - 7
______________
x - 1
The expression on being divide by x-1 gives -5 as remainder and quotient (x²-x+2)
What is a polynomial expression?Polynomial is made up of two terms, namely Poly (meaning “many”) and Nominal (meaning “terms.”). A polynomial is an expression composed of variables, constants, and exponents, combined using mathematical operations such as addition, subtraction, multiplication, and division (No division operation by a variable). Based on the number of terms present in the expression, it is classified as monomial, binomial, and trinomial. For example P(x) = x²-5x+11
Given here: The expression as x³-2x²+3x-7 which is to be divide by x-1
Thus dividing using the long divsion method we get
x-1)x³-2x²+3x-7(x²-x+2
x³-x²
-________
0-x²+3x
x²+x
-________
2x-7
2x-2
-__________
-5
Thus The expression can be written as x³-2x²+3x-7 =(x-1)×(x²-x+2)-5
Hence, On dividing the given expression by x-1 by long divison method we get the remainder as 2.
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which of the following is an example of within-subject variable?
Answer:
A within-subject variable is a type of variable in which the same subjects are used to measure multiple levels or conditions of the variable. One example of a within-subject variable would be measuring a person's reaction time to a stimulus both before and after a training intervention.
For example: measuring a person's reaction time to a visual stimulus before and after a training program in which the participant learns to react faster to the stimulus, here reaction time is the within-subject variable.
In order to set premiums at profitable levels, insurance companies must estimate how much they will to pay in claims on cars of each make and model, based on the value of the car and how much damage it sustains in accidents. Let C be a random variable that represents the cost of a randomly selected car of one model to the insurance company. The probability distribution of C is given below C $0 $1000 $4000 $10,000 P(C) 0.60 0.05 0.13 0.22 The probability that the insurance company will have to pay a claim of at least $4000 for a randomly selected car is: a.0.05 b.0.13. c.0.22. d.0.35. e.0.406. The expected value of C is a.$0 b.$554 c.$2770 d.$3750 e.$40577. Which of the following is the best interpretation of expected value? (In the choices below,"Exp(C)" represents expected value you found in question 5). a.If the company insures 10 cars of this model, they know they will incur 10×Exp(C) in costs. b.The maximum cost to the company for insuring this car model is Exp(C) per car. c.The company must insure at least Exp(C) of these cars to make a profit. d.If the company insures a large number of these cars, they canexpect the cost per car to average approximately Exp(C). e.If the company insures a large number of these cars, they can expect the variability in cost per car to average approximately Exp(C).
The probability of a claim of at least $4000 is 0.35, option(d), the expected value of C is $2770 option (c), and the best interpretation is If the company insures a large number of these cars, they can expect the cost per car to average approximately Exp(C), option(d).
Given probability distribution
C $0 $1000 $4000 $10000
P(C) 0.60 0.05 0.13 0.22
The probability that the insurance company will have to pay a claim of at least $4000 for a randomly selected car is
= (1-0.60-0.05) = 0.35
so the expected value of C
= 0x0.60 + 1000x0.05 + 4000x0.13 + 10000x0.22
= 50+520+2200 = $2770
Now, the best interpretation of expected value:
If the company insures a large number of these cars, they can expect the cost per car to average approximately Exp(C).
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The weight of an organ in adult males has a bell-shaped distribution with a mean of 330 grams and a
standard deviation of 50 grams. Use the empirical rule to determine the following.
(a) About 95% of organs will be between what weights?
(b) What percentage of organs weighs between 280 grams and 380 grams?
(c) What percentage of organs weighs less than 280 grams or more than 380 grams?
(d) What percentage of organs weighs between 180 grams and 430 grams?
According to the empirical rule -
a) About 95% of organs will be in range of weights of organs 230 grams and 430 grams.
b) 68% percentage of organs weighs between 280 grams and 380 grams.
c) 32% percentage of organs weighs less than 280 grams or more than 380 grams.
d) 97.35% percentage of organs weighs between 180 grams and 430 grams.
What is empirical rule?
The empirical rule 68-95-99.7 tells us that:
68% of the data is expected to be within 1 standard deviation from the mean.
95% of the data is expected to be within 2 standard deviation from the mean.
99.7% of the data is expected to be within 3 standard deviation from the mean.
There is a bell shaped distribution ( assume as approximately normal) with mean of 330 g and standard deviation of 50 g.
a) This happens for an interval with ±2 standard deviations from the mean.
That is:
X1 = μ + z1 × σ X2 = μ + z2 × σ
X1 = 330 - 2 × 50 X2 = 330 + 2 × 50
X1 = 330 - 100 X2 = 330 + 100
X1 = 230 X2 = 430
Therefore, the range of weights of organs is 230 grams and 430 grams.
b) Calculate the z-scores for each value to know how many standard deviations are from the mean.
z1 = (x1 - μ)/σ z2 = (x2 - μ)/σ
z1 = (280 - 330)/50 z2 = (380 - 330)/50
z1 = -50/50 z2 = 50/50
z1 = -1 z2 = 1
The values are 1 standard deviations from the mean each.
Therefore, the expected percentage is 68%.
c) This is the complementary of the point b.
Then, it is expected that (100 - 68)% = 32% of the organs weigh less than 280 grams or more than 380 grams.
d) Calculate the z-scores for each value to know how many standard deviations are from the mean.
z1 = (x1 - μ)/σ z2 = (x2 - μ)/σ
z1 = (180 - 330)/50 z2 = (430 - 330)/50
z1 = -150/50 z2 = 100/50
z1 = -3 z2 = 2
The values are 1 standard deviations from the mean each.
Therefore, the expected percentage is 68%.
The values are 3 standard deviations from the mean each. Between 180 and 330 there is 99.7%/2 = 49.85% of the data.
The values are 2 standard deviations from the mean each. Between 330 and 430 there is 95%/2 = 47.5% of the data.
Then, between 180 grams and 430 grams there is (49.85% + 47.5)% = 97.35% of the data.
Therefore, the expected percentage is 97.35%.
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dillon places a ladder against the wall the base of the ladder is 5 feet from the wall the ladder is 12 feet long how will shorting the distance between the base of the ladder and the wall affect the dimentions of the triangle they form explain in terms of the pyhagoren therom
The base distance will be 10.9 feet .
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
The Pythagorean theorem states that, in a right triangle, the length of the hypotenuse squared equals the sum of the squares of the other two sides or legs of the triangle. Algebraically, a2+b2=c2. In your example, the length of the ladder would be the hypotenuse and the distance from the wall would be one of the legs. So substitute these values into the equation to get: 25+b2=144 or 25+b2=144 and then subtract 25 from both sides to get b2=119. Therefore b=√119≈10.908. If your problem requires an exact answer in its most simplified form, please note that √119=10.9
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In the supermarket, 7/12 of Marco’s bill is for fruit and vegetables. If 1/8 is for fruit, what fraction is for vegetables?
In other words, 3/4 fraction of the bill are for vegetables, while 1 part is for fruit.
How is this determined?Using the data provided in the issue, we can determine what percentage of Marco's bill is made up of veggies. We are aware that fruit and vegetables account for 7/12 of the total bill and make up 1/8 of the total. By deducting the portion of the bill that is for fruit from the portion that is for fruit and vegetables, we can determine the portion of the cost that is for vegetables.
We must identify a common denominator for the two fractions in order to accomplish this. Since 12 and 8 have a denominator of 24, we convert both fractions to have a denominator of 24.
1/8 turns becomes 3/24
7/12 changes to 21/24.
We deduct 3/24 from 21/24 in order to derive the percentage of fruit from the percentage of fruit and vegetables.
21/24 - 3/24 = 18/24
Vegetables make up 18/24, or 3/4, of Marco's bill.
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Multiply mentally 70 • 100 • 400 = (Type a whole number)
Use the theorems from Section 3.2 and the converses of those theorems in this section to write three biconditional statements about parallel lines and transversals.
The slopes of the two lines that form them must be equal (m1 = m2). This implies that the two lines are parallel.
1) Biconditional statement: If two lines are parallel, then the alternate interior angles formed by a transversal are equal.
Formula: m∠1 = m∠3 and m∠2 = m∠4
Calculation: If two lines are parallel with a transversal intersecting them, the slopes of the two lines must be equal (m1 = m2) and the alternate interior angles formed by the transversal must have the same measure (m∠1 = m∠3 and m∠2 = m∠4).
2) Biconditional statement: If two lines are parallel, then the corresponding angles formed by a transversal are equal.
Formula: m∠1 = m∠4 and m∠2 = m∠3
Calculation: If two lines are parallel with a transversal intersecting them, the slopes of the two lines must be equal (m1 = m2) and the corresponding angles formed by the transversal must have the same measure (m∠1 = m∠4 and m∠2 = m∠3).
3) Biconditional statement: If two angles are equal, then the lines that form them are parallel.
Formula: m1 = m2
Calculation: If two angles have the same measure, then the slopes of the two lines that form them must be equal (m1 = m2). This implies that the two lines are parallel.
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