This is a contradiction, which means that there is no solution for the given equation. Therefore, the correct answer is option C, "No solution".
There is only one solution for the given equation.
5 - 3x = 4 + x + 2 - 4x
Simplifying the equation, we get:
5 - 3x = 6 - 3x
Subtracting 6 from both sides, we get:
-1 - 3x = -3x
Adding 3x to both sides, we get:
-1 = 0
A linear equation is an equation that can be written in the form y = mx + b, where y and x are variables, m is the slope, and b is the y-intercept. It represents a straight line on a graph. Linear equations can be used to model a variety of real-world situations, such as the relationship between temperature and time, or the cost of producing a certain quantity of goods.
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We can see that both sides are equal, which means that the equation has infinitely many solutions.
Therefore, the answer is: D. Infinitely many solutions is correct.
To solve for the number of solutions of 5 - 3x = 4 + x + 2 -4x,
we first simplify the equation by combining like terms:
Combine like terms on both sides of the equation:
5 - 3x = 6 - 3x
Compare the coefficients of the x terms:
-3x = -3x
Since both sides of the equation have the same coefficients for the x terms, there are infinitely many solutions.
Therefore, the answer is: D. Infinitely many solutions is correct.
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Can someone answer this please and thank you.
The blue base is the face (put in 1).
The black line is the edge (put in 2).
The dot up top is the vertex (put in 3).
Write the functions in standard form:
h(x)=2(x-3)²-9
h(x)=
p(x) = -5(x + 2)² + 15
p(x)=
Answer:
[tex]h(x)=2x^2-12x+9[/tex], [tex]p(x)=-5x^2-20x-5[/tex]
Step-by-step explanation:
To get to the standard form of a quadratic equation, we need to expand and simplify. Recall that standard form is written like so:
[tex]ax^2+bx+c[/tex]
Where a, b, and c are constants.
Let's expand and simplify h(x).
[tex]2(x-3)^2-9=\\2(x^2+9-6x)-9=\\2x^2+18-12x-9=\\2x^2+9-12x=\\2x^2-12x+9[/tex]
Thus, [tex]h(x)=2x^2-12x+9[/tex]
Let's do the same for p(x).
[tex]-5(x+2)^2+15=\\-5(x^2+4+4x)+15=\\-5x^2-20-20x+15=\\-5x^2-5-20x=\\-5x^2-20x-5[/tex]
Thus, [tex]p(x)=-5x^2-20x-5[/tex]
Slope-intercept (0, -2) , (9,1)
dora drove east at a constant rate of 75 kph. one hour later, tim started driving on the same road at a constant rate of 90 kph. for how long was tim driving, before he caught up to dora? a. 5 hours b. 4 hours c. 3 hours d. 2 hours
Tim was driving for 5 hours before he caught up to Dora.
The answer is (a) 5 hours.
To solve this problem, we can use the formula:
distance = rate × time
Let's denote the time Tim drove as t hours.
Since Dora started driving one hour earlier, her driving time would be (t + 1) hours.
Dora's distance: 75 kph × (t + 1)
Tim's distance: 90 kph × t
Since Tim catches up to Dora, their distances will be equal:
75(t + 1) = 90t
Now we can solve for t:
75t + 75 = 90t
75 = 15t
t = 5.
The answer is (a) 5 hours.
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Using the graph, determine the coordinates of the x-intercepts of the parabola.
Answer:
x = -5, x = 1
As (x, y) coordinates, the x-intercepts are (-5, 0) and (1, 0).
Step-by-step explanation:
The x-intercepts are the x-values of the points at which the curve crosses the x-axis, so when y = 0.
From inspection of the given graph, we can see that the parabola crosses the x-axis at x = -5 and x = 1.
Therefore, the x-intercepts of the parabola are:
x = -5x = 1As (x, y) coordinates, the x-intercepts are (-5, 0) and (1, 0).
please help me find the answer!! this is due tmmr!!
Answer:
Step-by-step explanation:
a coin is tossed 10,000 times. what is the chance that the number of heads will be in the range 4850 to 5150?
The chance that the number of heads will be in the range 4850 to 5150 is approximately 0.9973, or about 99.73%.
The number of heads in 10,000 coin tosses follows a binomial distribution with parameters n = 10,000 (the number of trials) and p = 0.5 (the probability of heads on a single toss).
We can approximate this binomial distribution using the normal distribution, with mean μ = np = 5000 and variance σ² = np(1-p) = 2500.
To find the probability that the number of heads is in the range 4850 to 5150, we can use the normal distribution and standardize the range using the z-score formula:
z = (x - μ) / σ
where x is the number of heads in the range we're interested in.
For the lower bound of 4850, we have:
[tex]z_lower = (4850 - 5000) / \sqrt{(2500)}[/tex]
= -3
For the upper bound of 5150, we have:
[tex]z_upper = (5150 - 5000) / \sqrt{(2500)} = 3[/tex]
Using a standard normal distribution table or calculator, we can find the probability of being within 3 standard deviations of the mean:
P([tex]z_lower[/tex] < Z < [tex]z_upper[/tex] ) ≈ P(-3 < Z < 3)
= 0.9973.
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what is 72% written in a deciamal
Eddie Clauer sells a wide variety of outdoor equipment and clothing. The company sells both through mail order and via the internet. Random samples of sales receipts were studied for mail-order sales and internet sales, with the total purchase being recorded for each sale. A random sample of 17 sales receipts for mail-order sales results in a mean sale amount of $84. 80 with a standard deviation of $19. 25. A random sample of 12 sales receipts for internet sales results in a mean sale amount of $77. 10 with a standard deviation of $26. 25. Using this data, find the 90% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases. Assume that the population variances are not equal and that the two populations are normally distributed.
Step 1 of 3 :
Find the critical value that should be used in constructing the confidence interval. Round your answer to three decimal places.
Step 2 of 3
Find the Staandard error of the sampling distrbution to be used in constructing the confidence interval
Step 3 of 3
you were to ask to construct the 90% confidence interval, given the following information
The 90% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases is approximately [-6.62, 22.02].
The critical value that should be used in constructing the confidence interval.
Since we are looking for a 90% confidence interval, we need to find the critical value associated with a 5% level of significance in a two-tailed test.
Using a t-distribution with (n1-1) + (n2-1) degrees of freedom and a significance level of 0.05, we find the critical value to be:
t-critical = 1.717 (using a t-distribution table or a calculator)
Step 2 of 3:
Next, we need to find the standard error of the sampling distribution to be used in constructing the confidence interval.
Since the population variances are not equal, we need to use the Welch-Satterthwaite equation to calculate the standard error:
SE = sqrt[([tex]s1^2[/tex]/n1) + ([tex]s2^2[/tex]/n2)]
where s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
Substituting the given values, we get:
SE = sqrt[([tex]19.25^2[/tex]/17) + ([tex]26.25^2[/tex]/12)]
SE ≈ 8.35
Step 3 of 3:
To construct the 90% confidence interval, we can use the formula:
(mean1 - mean2) ± t-critical * SE
where mean1 and mean2 are the sample means, and t-critical and SE are the values calculated in steps 1 and 2.
Substituting the given values, we get:
= (84.80 - 77.10) ± 1.717 x 8.35
= 7.70 ± 14.32
Therefore,
The 90% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases is (approx) [-6.62, 22.02].
We can be 90% confident that the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases falls within this interval.
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PLEASE HELP AND EXPLAIN AND SHOW WORK ON HOW YOU GOT THE ANSWER I WILL MARK YOU BRAINLIEST. PLEASE EXPLAIN HOW YOU GOT THE ANSWER!!!
The terms arranged in order from smallest to biggest are: (-2)³, -√25, √11, 10, and 4² after comparing the values of the final numbers.
How to arrange the terms of numbers in ascending orderWe shall first simplify the numbers to get their final values and then compare to which is smaller as follows:
4² = 4 × 4 = 16
-√25 = -5
10 = 10
√11 = 3.3166
(-2)³ = -2 × -2 × -2 = -8
In conclusion, we have by comparing the final values of the numbers the terms arranged from smallest to the biggest as: (-2)³, -√25, √11, 10, and 4².
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a plane travels 600 from salt lake city, utah, to oakland, california, with a prevailing wind of 30. the return trip against the wind takes longer. find the average speed of the plane in still air.
the average speed of the plane in still air is s + 30.
Let's call the average speed of the plane in still air "s" (in miles per hour).
We can use the formula:
time = distance / speed
to find the time it takes the plane to travel from Salt Lake City to Oakland with the wind and against the wind.
With the wind:
time with wind = [tex]600 / (s + 30)[/tex]
Against the wind:
time against wind =[tex]600 / (s - 30)[/tex]
time against wind > time with wind
So we can set up an inequality:
[tex]600 / (s - 30) > 600 / (s + 30)[/tex]
Multiplying both sides by [tex](s - 30)(s + 30)[/tex], we get:
[tex]600(s + 30) > 600(s - 30)[/tex]
Expanding and simplifying, we get:
[tex]600s + 18000 > 600s - 18000[/tex]
Subtracting 600s from both sides, we get:
[tex]18000 > -18000[/tex]
This inequality is true for all values of s. In other words, there are no restrictions on the value of s that would make the return trip take longer than the trip with the wind.
Therefore, we can use the average of the two speeds (with and against the wind) to find the average speed of the plane in still air:
Average speed = [tex]2s(s + 30) / (s + 30 + s - 30)[/tex]
Simplifying, we get:
Average speed = [tex]2s(s + 30) / (2s)[/tex]
Canceling the common factor of 2s, we get:
Average speed = s + 30
We know that the distance from Salt Lake City to Oakland is 600 miles, and we can use the formula:
time = distance / speed
to find the time it takes the plane to travel this distance:
time = [tex]600 / (s + 30)[/tex]
We also know that the return trip (against the wind) takes longer, so we can set up another equation:
time return trip =[tex]600 / (s - 30)[/tex]
We can use these two equations to solve for s:
[tex]600 / (s + 30) = 600 / (s - 30)[/tex]
Cross-multiplying, we get:
[tex]600(s - 30) = 600(s + 30)[/tex]
Expanding and simplifying, we get:
[tex]600s - 18000 = 600s + 18000[/tex]
Subtracting 600s from both sides, we get:
[tex]-18000 = 18000[/tex]
This is not a valid equation, so there must be no solution.
However, we can still find the average speed of the plane in still air by using the equation we derived earlier:
Average speed = s + 30
So the average speed of the plane in still air is s + 30. We don't have a specific value for s, but we can say that the average speed is equal to the speed with the wind plus 30 (which is the speed of the wind).
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Which expressions are equivalent to 27^4/3?
Select the three correct answers.
A. 4^3
B. (27^1/3)^4
C. 3^1/4
D. 81
D) 81 is equivalent to 27^(4/3).
The expression 27^4/3 can be simplified using the rule that (a^m)^n = a^(m*n). Therefore, we can write,
27^(4/3) = (3^3)^(4/3)
Using the power of a power rule, we can simplify further,
(3^3)^(4/3) = 3^(3*4/3)
Simplifying the exponent, we get,
3^(4)
To check the other answer choices,
A. 4^3 is not equivalent to 27^4/3.
B. (27^1/3)^4 is equivalent to 27^(4/3), which we already simplified to 3^4. Therefore, this expression is also equivalent to 3^4.
C. 3^1/4 is not equivalent to 27^4/3.
D. 81 is equivalent to 3^(4).
Therefore, the expression 27^4/3 is equivalent to 3^4, which is answer choice D) 81.
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ehat are the roots of the polynominal equation? use a grapghing calculator and make 0=y,and find the x intercepts. x2 + x - 72=0 enter you answers in the boxes.
Therefore, the roots of the polynomial equation x² + x - 72 = 0 are -9 and 8.
What is quadratic equation?A quadratic equation is a type of polynomial equation of the second degree, which means it has one or more terms in which the variable is raised to the power of two, but no higher powers.Quadratic equations can have zero, one, or two real solutions, depending on the values of a, b, and c. These solutions are also called the roots or zeros of the equation.
Here,
To find the roots of the polynomial equation x² + x - 72 = 0, we can set y = 0 and solve for x. This is equivalent to finding the x-intercepts of the graph of the function f(x) = x² + x - 72.
We can use the quadratic formula to solve for x:
x = (-b ± √(b² - 4ac)) / (2a)
where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
In this case, a = 1, b = 1, and c = -72, so we have:
x = (-1 ± √(1² - 4(1)(-72))) / (2(1))
x = (-1 ± √(1 + 288)) / 2
x = (-1 ± √(289)) / 2
x = (-1 ± 17) / 2
Therefore, the roots of the polynomial equation x² + x - 72 = 0 are:
x = -9 or x = 8
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erin is playing darts at the adventure arcade. she scores a bullseye 15% of the time, and she is about to throw 5 darts. how likely is it that she will get at least one bullseye?
the likelihood of Erin getting at least one bullseye in 5 throws is 0.5563 or 55.63%.
To calculate the likelihood of Erin getting at least one bullseye, we need to first calculate the probability of her not getting a bullseye in a single throw. Since she scores a bullseye 15% of the time, the probability of her not getting a bullseye in a single throw is 85% (100% - 15%).
Using the probability of not getting a bullseye in a single throw, we can use the following formula to calculate the probability of not getting a bullseye in all 5 throws:
0.85 x 0.85 x 0.85 x 0.85 x 0.85 = 0.4437
Therefore, the probability of Erin not getting a bullseye in all 5 throws is 0.4437 or 44.37%.
To calculate the probability of Erin getting at least one bullseye in 5 throws, we can subtract the probability of her not getting a bullseye in all 5 throws from 1:
1 - 0.4437 = 0.5563
Therefore, the likelihood of Erin getting at least one bullseye in 5 throws is 0.5563 or 55.63%.
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The probability that Erin will get at least one bullseye in her 5 throws at the adventure arcade is approximately 55.63%.
To find the probability that she will get at least one bullseye in 5 throws, we can use the complementary probability.
This means we will first find the probability of her not getting a bullseye in all 5 throws, and then subtract that from 1.
Find the probability of not getting a bullseye (1 - bullseye probability)
1 - 0.15 = 0.85
Calculate the probability of not getting a bullseye in all 5 throws
0.85^5 ≈ 0.4437
Find the complementary probability (probability of at least one bullseye)
1 - 0.4437 ≈ 0.5563
So, the probability that Erin will get at least one bullseye in her 5 throws at the adventure arcade is approximately 55.63%.
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Write your answer as an integer or decimal.
please help
The value of angle GFH is 18°
What is circle geometry?A circle is a special kind of ellipse in which the eccentricity is zero and the two foci are coincident.
A theorem in circle geometry starts that angle in the same segment are equal. In triangle EFG, angle F and G are on the same segment, this means that angle F and G are equal.
Represent angle F as x
therefore 144+2x = 180° ( sum of angle in a triangle)
2x = 180-144
2x = 36
x = 36/2 = 18°
Therefore the measure of angle GFH is 18°
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I need some help pretty please
Answer:
Step-by-step explanation:
Change in Y/Change in X
5- -1/6- -2
5+1/6+2
6/8
3/4
I need help please I will give brainliest to the best answer...
The value of x in the intersecting chords that extend outside circle is 5
Calculating the value of xFrom the question, we have the following parameters that can be used in our computation:
intersecting chords that extend outside circle
Using the theorem of intersecting chords, we have
4 * (x + 6 + 4) = 6 * (x - 1 + 6)
Evaluate the like terms
So, we have
4 * (x + 10) = 6 * (x + 5)
Using a graphing tool, we have
x = 5
Hence. the value of x is 5
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please solve correctly my grade depends on it
Just use the pythagorean theorem to solve the hypotenuse!
(3^2)+(2^2)=x^2
9+4=13^2
[tex]\sqrt{13}[/tex] = [tex]\sqrt{x}[/tex]
[tex]13^{2}[/tex] km
Hope this helps <3
Help please? I just need an answer. A clear explanation earns brainliest.
the simplified form of expression is: -(x² + 2x - 2)/((x+2)*(x+4))
what is expression ?
In mathematics, an expression is a combination of numbers, variables, operators, and/or functions that represents a mathematical quantity or relationship. Expressions can be simple or complex
In the given question,
To evaluate the expression 1/(x+2) - (x+1)/(x+4), we need to find a common denominator for the two terms. The least common multiple of (x+2) and (x+4) is (x+2)(x+4).
So, we can rewrite the expression as:
(1*(x+4) - (x+1)(x+2))/((x+2)(x+4))
Expanding the brackets, we get:
(x+4 - x² - 3x - 2)/((x+2)*(x+4))
Simplifying the numerator, we get:
(-x² - 2x + 2)/((x+2)*(x+4))
Therefore, the simplified expression is:
-(x² + 2x - 2)/((x+2)*(x+4))
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Find unknown sides and angle of the triangle
The sides and the angle of the right triangle are a = 10√2, b = 10√2 and B = π / 4.
How to find the missing information of a right triangle
In this problem we need to determine the values of two sides and an angle of the right triangle. This can be done by means of the following properties:
A + B + C = π
sin A = a / c
cos A = b / c
tan A = a / b
Where:
A, B, C - Angles of the right triangle, in radians.a, b, c - Sides of the right triangle.If we know that A = π / 4, C = π / 2 and c = 20, then the missing angle and missing sides are, respectively:
B = π - π / 4 - π / 2
B = π / 4
cos (π / 4) = b / 20
b = 20 · cos (π / 4)
b = 10√2
sin (π / 4) = a / 20
a = 20 · sin (π / 4)
a = 10√2
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Five balls, A, B, C, D, and E, weigh 30g, 50g, 50g, 50g, and 80g each. Which ball weighs 30g?
The ball that weighs 30g is ball A.
What is accuracy and precision?The degree to which a measured value resembles the true or recognised value is known as accuracy. On the other hand, precision describes how closely two measurements of the same quantity agree. In other words, precision is a measure of consistency, whereas accuracy is a measure of correctness. The values obtained are consistent but not always close to the genuine value when a measurement is precise but not exact. On the other hand, a measurement might be accurate without being exact, which means that the result is close to the correct value but produces erratic or dispersed results when repeated.
From the given weights corresponding to different balls, 30 g corresponds to ball A.
Hence, the ball that weighs 30g is ball A.
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The solid below is dilated by a scale factor of 1/2. Find the volume of the
solid created upon dilation.
24
26
10
34
Answer: 4080
Step-by-step explanation:
First you have to find the area of the triangle. 24*10 = 240. 240/2 = 120. Then you multiply the area of the triangle and multiply it by 34. 120 * 34 = 4080. This means the answer is 4080
a p-value a. can be positive or negative. b. is a probability. c. can be smaller than 0 but no larger than 1. d. can be larger than 1 but no smaller than 0. e. can only range in value from -1 to 1.
A p-value is a probability.
A p-value is the probability of obtaining a test statistic as extreme or more extreme.
The observed value, assuming the null hypothesis is true.
It ranges in value from 0 to 1 and represents the strength of evidence against the null hypothesis.
A p-value cannot be negative, as it is a probability and probabilities are always between 0 and 1.
A p-value also cannot be larger than 1, as it represents a probability.
A probability cannot exceed 1.
Finally, a p-value cannot be smaller than 0, as it represents a probability.
A probability cannot be negative.
the correct option is b. is a probability.
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Determine whether ▰ABCD with vertices A(-4,6), B(-1,7), C(0,4), and D(-3,3) is a rhombus, a rectangle, a square, or none. Select all the apply.
~a.) Rhombus
~b.) Rectangle
~c.) Square
~d.) None
The only statement that is true is b, which states that the quadrilateral is a rectangle.
What is quadrilateral?A quadrilateral is a polygon with four sides and four vertices. The sum of the interior angles of a quadrilateral is always 360 degrees. Quadrilaterals can have sides of different lengths and angles of different measures, giving rise to many different types of quadrilaterals with different properties.
According to the given informationFirst, we find the lengths of the sides of the quadrilateral:
AB = √[(7-6)² + (-1+4)²] = √10
BC = √[(4-7)² + (0-0)²] = 3
CD = √[(3-4)² + (-3+0)²] = √10
AD = √[(6-3)² + (-4+1)²] = √26
Then, we find the slopes of each pair of opposite sides:
AB: (7-6)/(−1+4) = 1/3
BC: (4-0)/(0-(-1)) = 4/1 = 4
CD: (-3-(-4))/(0-(-3)) = 1/3
AD: (6-3)/(-4-(-1)) = -1/5
Now we can analyze each statement:
a.) Rhombus
A rhombus is a quadrilateral with all sides of equal length. We found that AB = CD and AD ≠ BC, so not all sides are of equal length. Therefore, statement a is false.
b.) Rectangle
A rectangle is a quadrilateral with all angles equal to 90 degrees. We can find the slopes of adjacent sides and check if they are opposite reciprocals:
AB: 1/3
BC: 4
CD: 1/3
AD: -1/5
We can see that AB and CD have slopes of 1/3 and are opposite reciprocals, and BC and AD have slopes of 4 and -1/5, respectively, and are also opposite reciprocals. Therefore, all angles of the quadrilateral are 90 degrees. Also, since AB = CD and AD ≠ BC, the quadrilateral is a rectangle. Therefore, statement b is true.
c.) Square
A square is a special type of rectangle with all sides of equal length. We found that AB ≠ AD, so not all sides are of equal length. Therefore, statement c is false.
d.) None
We have determined that the quadrilateral is a rectangle, so it is not "none". Therefore, statement d is false.
Therefore, the only statement that is true is b, which states that the quadrilateral is a rectangle.
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slove and answer x+y=11 2x-y=19
Answer:
x + y = 11
2x - y = 19
--------------
3x = 30
x = 10, so y = 1
40000 is divided by the smallest number so that the result is a perfect cube. find the cube root of the resulting number.
The Cube root of the resulting number is 8.
The smallest number that 40000 can be divided by so that the result is a perfect cube, we need to factorize 40000 into its prime factors:
[tex]40000 = 2^6 \times 5^4[/tex]
To make this a perfect cube, we need to ensure that the powers of each prime factor are multiples of 3.
The smallest number we can divide 40000 by so that the result is a perfect cube is:
[tex]40000 = 2^6 \times 5^4[/tex]
Now we can find the cube root of the resulting number:
[tex]3\sqrt (40000 \div 100) = 3\sqrt400 = 8.[/tex]
Factories 40000 into its prime components in order to determine.
The least number that the result may be divided by while still producing a perfect cube.
The powers of each prime factor must be multiples of three in order for this to be a perfect cube.
The least number that 40000 may be divided by to produce a perfect cube is:
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In the accompanying diagram, m<A=32° and AC = 10. Which equation could be used to find x in ∆ABC?
1. x=10 sin [32°]
2. x=10 cos [32°]
3. x = 10 tan [32°]
4. x=10/cos32
The equation x = 10 tan (32°) could be used to find x in ∆ABC.
RIGHT TRIANGLEA triangle is classified as a right triangle when it presents one of your angles equal to 90º. The greatest side of a right triangle is called the hypotenuse. And, the other two sides are called legs.
The math tools applied for finding angles or sides in a right triangle are the trigonometric ratios or the Pythagorean Theorem.
The Pythagorean Theorem says: (hypotenuse)²= (leg1)²+(leg2)² . And the main trigonometric ratios are: sin (x) , cos (x) and tan (x) , where:
[tex]sin(x)=\frac{opposite\ side}{hypotenuse} \\ \\ cos(x)=\frac{adjacent\ side}{hypotenuse}\\ \\ tan(x)=\frac{sin(x)}{cos(x)} =\frac{opposite\ side}{adjacent\ side}[/tex]
The question gives the value of the two sides and the value of an angle. From the trigonometric ratios presented before, you can write:
[tex]tan(32)=\frac{opposite\ side}{adjacent\ side}=\frac{x}{10} \\ \\ x=10\ tan (32\°)[/tex]
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To determine whether 2126.5
and 58158
are in a proportional relationship, write each ratio as a fraction in simplest form.
What is 2126.5
as a fraction in simplest form?
Enter your answer in the box.
Answer:
both are 5/13the relationship is proportionalStep-by-step explanation:
You want to know if the fractions (2 1/2)/(6.5) and (5/8)/(1 5/8) are in a proportional relationship, and the simplest form of each.
FractionsEquivalent fractions can be found by multiplying numerator and denominator by the same number.
(2 1/2)/(6.5) = 2·(2 1/2)/(2·6.5) = 5/13
(5/8)/(1 5/8) = 8(5/8)/(8·(1 5/8)) = 5/(8+5) = 5/13
Both fractions are equivalent to 5/13, so their relationship is proportional.
Find the points on the surface z2 = xy +16 closest to the origin. The points on the surface closest to the origin are (Type an ordered triple. Use a comma to separate answers as needed. )
The points on the surface z² = xy + 16 closest to the origin are: (-4,4,0) and (4, -4, 0)
We know that the distance between an arbitrary point on the surface and the origin is d(x, y, z) = √(x² + y² + z²)
Using Lagrange multipliers,
L(x, y, z, λ) = x² + y² + z² + λ(z² - xy - 16)
We have partial derivatives.
[tex]L_x[/tex] = 2x - λy
[tex]L_y[/tex] = 2y - λx
[tex]L_z[/tex] = 2z + 2zλ
[tex]L_\lambda[/tex] = z² - xy - 16
Now we set each partial derivative to zero to find critical points.
[tex]L_x[/tex] = 0
2x - λy = 0
[tex]L_y[/tex] = 0
2y - λx = 0
After solving above equations simultaneously we get (x + y)(x - y) = 0
i.e., x = -y OR x = y
[tex]L_z[/tex] = 0
2z + 2zλ = 0
z = 0 OR λ = 0
Consider [tex]L_\lambda[/tex] = 0
z² - xy - 16 = 0
-xy = 16 ............(as z = 0)
when x = y then -y² = 16 which is not true.
So, consider x = -y
-(-y)y = 16
y² = 16
y = ±4
when y = 4 then we get x = -4
and when y = -4 then we get x = 4
Therefore, the closest points are:(-4,4,0) and (4, -4, 0)
Learn more about the Lagrange multipliers here:
https://brainly.com/question/30776684
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Round the number. Write the result as the product of a single digit and a power of 10.
4,241,933,200