The explicit formula for the geometric series is [tex]a_n = 3(5)^{n-1}[/tex].
What is geometric progression?A geometric series is one in which there is a constant separating the two succeeding numbers. In other words, to create a geometric progression, each succeeding term is multiplied by the constant term.
Given that the geometric sequence is,
3, 15, 75......
The common ratio of the series is calculated as the ratio of the second term to the first term,
r = 15 / 3
r = 5
The explicit formula can be written as:-
[tex]a_n = a_1(r)^{n-1}[/tex]
[tex]a_n = 3(5)^{n-1}[/tex]
Therefore, the formula is [tex]a_n = 3(5)^{n-1}[/tex].
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a person eating at a cafeteria must choose 3 of the 19 vegetables on offer. calculate the number of elements in the sample space for this experiment.
The number of elements in the sample space for this experiment are 969 by combination formula.
In an experiment where a person eating at a cafeteria choose 3 out of the 19 vegetables on offer, the number of elements in the sample space is 969.
A sample space of an experiment is the set of all possible outcomes of a random experiment. So the number of elements in a sample space is the total number of possible outcomes of the experiment.
Here the experiment is choosing 3 vegetables from a set of 19 vegetables offered. The total number of ways to choose 3 out of 19 distinct vegetables = [tex]^{19}C_{3}[/tex]
= [tex]\frac{19!}{3!\times16!}[/tex]
= 969
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Which equation represents a parabola with a focus of (4,-3) and a directrix of y = 1?
The equation of the parabola is x^2 = 8(y + 3)
How to determine the equation of the parabolaFrom the question, we have the following parameters that can be used in our computation:
Focus = (4, -3)
Directrix, y = 1
Since the directrix is a horizontal line, the parabola opens downward or upward.
The focus is located 4 units to the right of the vertex
So, we have
Vertex = (0, -3).
The equation is of the form:
(x - h)^2 = 4p(y - k),
where (h, k) is the vertex, and p is the distance from the vertex to the focus (and to the directrix).
In this case, h = 0, k = -3, and p = 2.
Substituting these values, we get:
(x - 0)^2 = 8(y + 3)
x^2 = 8(y + 3)
So the equation is x^2 = 8(y + 3)
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39 sit-ups in 3 days
day
=
sit-ups per
Answer:
13 situps per day
Step-by-step explanation:
To find the number of situps per day, take the total number of situps and divide by the number of days.
39/3
13 situps per day
Type the correct answer in each box. Use numerals instead of words.
Solve the equation by completing the square. x2 + 20x + 82 = 7
The solutions, in order from least to greatest, are x = and x = .
By completing the square, the equation. x2 + 20x + 82 = 7. The solutions are x = -15 and x = -5, going from least to largest.
What is meant by constant?In expression, a constant is a value or number that is consistently the same. Multiple meanings can be derived from the word "constant." Its meanings as a noun and an adjective, respectively, are non-variance and two: an unchanging integer or other non-changing mathematical entity that is fixed and well-defined. The value of a constant is fixed and does not alter over time. A shoe's, a piece of clothing's, or any other item of clothing's size, for instance, will never alter. Because 8 is a constant number and cannot be modified in an algebraic equation, x+y = 8 is a valid solution.The square of half the x-coefficient must equal the constant on the left:
(20/2)^2 = 100
To get it to that value, we can add 18 to both sides of the equation:
x^2 + 20x + 100 = 25 . . . add 18 to both sides of the equation
(x + 10)^2 = 25 . . . . . . . . write the left side as a square
x + 10 = ±√25 = ±5 . . . . . take the square root
x = - 10 ± 5 = {-15, -5}
The solutions are x = -15 and x = -5.
The attached graph shows the solutions to ...
x^2 + 20x + 82 -7 = 0 . . . . . the result of subtracting 7 from both sides
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Shape A is reflected in the line with equation x = 2 to give Shape B
In mathematical terms, each point (x, y) on Shape A is mapped to a point (2 + (2 - x), y) on Shape B as reflected line equation and explained as
When a shape is reflected in a line, it means that each point on the original shape is mapped to a corresponding point on the reflected shape that is symmetrical with respect to the reflecting line.
The line of reflection can be any straight line, and in this case, the line of reflection is given by the equation x = 2. To understand how the reflection works, imagine holding a mirror along the line x = 2 and placing Shape A on the mirror. The reflected shape, Shape B, would be the image that you see in the mirror. In mathematical terms, each point (x, y) on Shape A is mapped to a point (2 + (2 - x), y) on Shape B. This means that for each point on Shape A, we take its x-coordinate, subtract it from 2 to find the distance from the line of reflection, and then add that distance to 2 to find the x-coordinate of the corresponding point on Shape B. The y-coordinate remains the same.
The line of reflection acts as a dividing line between Shape A and Shape B, and the points on Shape B are exactly the same distance from the line of reflection as the points on Shape A, but on the opposite side. This means that Shape B is a perfect mirror image of Shape A, reflected across the line x = 2.
In summary, the reflection of Shape A in the line x = 2 maps each point on Shape A to a corresponding point on Shape B that is symmetrical with respect to the line of reflection. The result is a perfect mirror image of the original shape, reflected across the line.
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Ken bought a bike that was priced at $450. He was offered a 25% discount what was the discount on the bike
If discount offered is 25% then the discount on the bike is $112.50 by multiply the rate by the original price.
What is discount price?
Discount pricing is a type of promotional pricing strategy where the original price for a product or service is reduced with the aim of increasing traffic, moving inventory, and driving sales.
According to the question:
Ken bought a bike that was priced at $450
Discount percentage offered to him is 25%
To find the discount price, first the rate is usually given as a percent.
To find the discount, multiply the rate by the original price.
Discount = [tex]450 \times \frac{25}{100}[/tex]
Therefore the discount on the bike is $112.50
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The cellular phone service for a business executive is $35 per month plus $0.40 per minute of phone use over 900 min. For a month in which the executive's cellular phone bill was $104.20, how many minutes did the executive use the phone?
Answer:1073
Step-by-step explanation: 104.2-35+69.2
69.2/.4=173
900+173+1073
Answer:
The exec used the phone for 1073 minutes
We just cancel out and isolate for x, then add the 900 minutes
Suppose you borrow $19,250 from a bank to purchase a house. You have poor credit score. If the term of the loan is 13 years, how much should you expect to pay in interest over the life of the loan if your interest rate is 10.50%?
A. 15,220
B. 35,360
C. 19,250
D. 16,110
$2627625 much should you expect to pay in interest over the life of the loan if your interest rate is 10.50%.
What is interest rate?The amount of interest payable each period expressed as a percentage of the amount lent, deposited, or borrowed is known as the interest rate. The total amount of interest on a loaned or borrowed sum is determined by the principle amount, the interest rate, the frequency of compounding, and the period of time during which it is lent, deposited, or borrowed.
Simple interest rate = P × R × T
where,
P = Principal = $19,250
R = Rate of Interest in % per annum = 10.50%
T = Time, usually calculated as the number of years = 13 years
Thus,
simple interest = 19250 × 10.50 × 13
simple interest = 2627625
$2627625 much should you expect to pay in interest over the life of the loan if your interest rate is 10.50%
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Han spent 60 minutes practicing the piano over the weekend. Priya practiced the violin for 75% as much time as Han practiced the piano. How long did she practice?
By using the unitary method to find out that Priya practiced the violin for 45 minutes, which is 75% of the time Han spent practicing the piano.
Unitary method is a method that involves finding the value of one unit and then using that value to find the value of multiple units. This method is particularly useful in solving problems that involve proportions, ratios, and rates.
Now let's apply the unitary method to the given problem. We know that Han practiced the piano for 60 minutes over the weekend. We also know that Priya practiced the violin for 75% as much time as Han practiced the piano. To find out how long Priya practiced, we need to first find out how long 75% of Han's practice time is.
To do this, we can use the following equation:
75% of Han's practice time = 75/100 x 60 minutes
= 45 minutes
So Priya practiced the violin for 45 minutes over the weekend.
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tyler drove 138miles, and brittney drove 116miles. what is the ratio of tyler's miles to brittney's miles? write your answer as a simplified fraction.
Tyler drove 138miles, and brittney drove 116miles, the ratio of tyler's miles to brittney's miles is 69/58.
The ratio of Tyler's miles to Brittney's miles can be found by dividing Tyler's miles by Brittney's miles.
Ratio of Tyler's miles to Brittney's miles = Tyler's miles / Brittney's miles
Ratio of Tyler's miles to Brittney's miles = 138 miles / 116 miles
Simplifying the ratio by dividing both numbers by their greatest common factor, which is 2, we get:
Hence,
Ratio of Tyler's miles to Brittney's miles = 69/58
This means that for every 69 miles Tyler drove, Brittney drove 58 miles.
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Of the following, which is not a solution to the differential equation y′′′+4y′=0?
A. Y=10
B. Y=4e^−2x
C. Y=3sin(2x)
D. Y=2cos(2x)+4
Answer:
A Y=10
Step-by-step explanation:
Answer:
Option B is the right answer.
Step-by-step explanation:
See Attachment
Suppose that a motorboat is moving at 83 ft/s when its motor suddenly quits, and that 1 s later the boat has slowed to 23 ft/s. Assume that the resistance it encounters while coasting is proportional to the square of its velocity so that dv/dt =-kv^2 where k > 0. How far will the boat coast in the first 2 minutes after its motor quits?
The boat travels a distance of [tex](1/2k) ln(61/42)^2[/tex] in the first 2 minutes after its motor quits.
We can solve this problem using separation of variables. The differential equation that models the velocity of the boat is:
[tex]dv/dt = -kv^2[/tex]
To do this, we first need to separate the variables and integrate both sides:
[tex]1/v^2 dv = -k dt[/tex]
Integrating both sides, we get:
-1/v = kt + C
where C is the constant of integration. To find C, we can use the initial condition that the boat is moving at 83 ft/s when the motor quits:
-1/83 = k(1) + C
C = -1/83 + k
Now we can substitute C back into our equation:
-1/v = kt - 1/83 + k
Solving for v, we get:
v = 1 / (k(t - 1/83 + 1))
Now we can integrate v over the time interval [1, 121] to find the distance traveled by the boat:
d = ∫[1,121] v dt
Substituting v into the integral, we get:
d = ∫[1,121] 1 / (k(t - 1/83 + 1)) dt
We can simplify the integral by using the substitution u = t - 1/83 + 1:
d = ∫[84/83, 122/83] 1 / (ku) du
Now we can integrate this expression to get:
d = ln(u) / k |[84/83, 122/83]
d = ln(122/83) / k - ln(84/83) / k
d = (1/k) ln(122/83) - (1/k) ln(84/83)
We can simplify this expression using the fact that C = -1/83 + k:
d = (1/k) ln(122/83 / 84/83)
d = (1/k) ln(61/42)
Finally, we can plug in the initial condition k > 0 to get the distance traveled by the boat:
d = [tex](1/k) ln(61/42) = (1/2k) ln(61/42)^2[/tex]
d = [tex](1/2k) ln(61/42)^2[/tex]
Since we are not given the value of k, we cannot compute the exact numerical value of the distance traveled.
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Match the information on the left with the appropriate equation on the right.
a) The equation of perpendicular line is y=2x+1
b) The equation of parallel line is y = -2x+6
What is slope ?
A line's steepness is determined by its slope. In mathematics, slope is determined by "rise over run" (change in y divided by change in x).
a) An equation perpendicular to y= (-1/2)x+4 through the point (-2, -3):
m= -1/2
Slope of perpendicular line is -1/m = -1/( -1/2) = 2
(x,y) = (-2, -3)
Slope intercept form:
y=mx+b
Substitute x,y and m values to find b
-3= 2(-2) + b
=> -3 = -4 + b
=> b= -3+4 = 1
So the equation of perpendicular line is y=2x+1
b) An equation through the point (1,4) parallel to y = -2x + 1
Parallel lines will have same slope , m= -2
(x,y) = (1,4)
Slope intercept form:
y=mx+b
Substitute x,y and m values to find b
4=-2(1)+b
=> 4 = -2 + b
=> b=4+2 = 6
So the equation of parallel line is y = -2x+6
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Someone please explain how to solve this.
Just multiply each side by side
Step-by-step explanation:1.Times one side by another
2. Times your answer by the third side
What is the sum of 9.72 × 108 and 1.93 × 107?
Answer:
The solution is 9,913 × 10⁸
Step-by-step explanation:
9,72 × 10⁸ and 1,93 × 10⁷
9,72 × 10⁸ + 1,93 × 10⁷
Factor the shape
= 10⁷ (9,72 × 10 + 1,93)
= 10⁷ (97,2 + 1,93)
= 10⁷ × 99,13
= 9,913 × 10⁸
the mayor of a town believes that above 72% of the residents favor annexation of an adjoining community. is there sufficient evidence at the 0.05 level to support the mayor's claim? state the null and alternative hypotheses for the above scenario.
Null hypothesis: Proportion of residents favoring annexation <= 72%. Alternative hypothesis: Proportion of residents favoring annexation > 72%
The null hypothesis is that the proportion of residents who favor annexation is equal to or less than 72%. The alternative hypothesis is that the proportion of residents who favor annexation is greater than 72%.
To determine if there is sufficient evidence to support the mayor's claim, a hypothesis test should be performed. A common approach is to use a one-sample proportion test. This test compares the proportion of residents who favor annexation in a sample to the hypothesized proportion of 0.72.
Assuming a significance level of 0.05, if the p-value is less than 0.05, then there is sufficient evidence to reject the null hypothesis and conclude that the proportion of residents who favor annexation is greater than 72%.
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ANSWER ATTACHED!
Enter a recursive rule for the geometric sequence. 10,−80,640,−5120....
P.s. If you're going to answer.. don't type random stuff, put the answer I gave you
The recursive rule for the geometric sequence 10, -80, 640, -5120, is a(1) = 10 (the first term) For n > 1, a(n) = -8*a(n-1) (each term is -8 times the previous term).
What does the term "recursive" in math mean?Recursive definitions of objects are those in which they are defined in terms of other objects of the same kind. The whole class of objects can then be built from a small set of initial values and a few of rules by utilising a recurrence relation of some sort. The most typical way to define the Fibonacci numbers is recursively.
geometric sequence 10, -80, 640, -5120
a(1) = 10 (the first term)
For n > 1, a(n) = -8*a(n-1) (each term is -8 times the previous term).
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Which expression is equivalent to (0.35t -0.4) - (0.8t - 0.35)?
Choose the correct answer below.
OA. 0.45t +0.75
OB. 0.45t -0.05
OC. 0.45t +0.75
OD.
0.45t -0.05
The equivalent expression is -0.45t - 0. 05. Option D
How to determine the equivalent expressionEquivalent expressions are those expressions having the same solution.
From the information given, we have the expression as;
(0.35t -0.4) - (0.8t - 0.35)
Now, to simply the expression, take the steps;
expand the bracket;
0.35t - 0.4 - 0.8t+ 0.35
Now, collect the like terms
0.35t - 0.8t - 0.4 + 0.35
Add or subtract the like terms
-0.45t - 0. 05
Hence, the expression is -0.45t - 0. 05
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Which composition of similarity transformations maps polygon ABCD to polygon A'B'C'D'?
The transformations map polygon ABCD to polygon A'B'C'D' shows dilation and then reflection.
What is transformation?Transform the shapes on a coordinate plane by rotating, reflecting, or translating them. In geometry, object transformations constitute the foundation for the majority of proofs.
The transformation, or f: X X, is the name given to a function, f, that maps to itself. After the transformation, the pre-image X becomes the picture X. Any operation, or a combination of operations, such as translation, rotation, reflection, and dilation, can be used in this transformation.
From the figure we observe that the figure is compressed or dilated using a negative scale factor, and then the figure is reflected on the y -axis.
Hence, the transformations map polygon ABCD to polygon A'B'C'D' shows dilation and then reflection.
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Ethan is planning for his retirement. He has narrowed it down to two investment options. The first is an IRA where monthly payments are made, in the amount of $416.66, for 30 years. The second is a Roth IRA where annual payments are made, in the amount of $5000, for 30 years. If both compound interest at a rate of 2.5%, determine which account will yield the largest future value for Ethan, and how much greater that value will be than that of the other account. Round your final answer to the nearest cent.
The account that will yield the greatest future value is the first IRA. The difference in value would be $3,403.07.
Which account will yield the greatest IRA?The formula for calculating future value = annuity factor x monthly payments
Annuity factor = {[(1+r)^n] - 1} / r
Where:
r = interest raten = number of periodsFuture value with monthly compounding: $416.66 x [{1.00208^360] - 1 }/0.00208] = $222,916.59
r = monthly interest rate = annual interest rate / 12 = 2.5% /12 = 0.208%n = number of periods = number of years x rate of compounding = 30 x 12 = 360Future value with annual compounding: 5000 x [{1.025^30) - 1} / 0.0205] = $219,513.52
Difference in values = $222,916.59 - $219,513.52 = $3,403.07
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you leave your house, travel one mile due south, then one mile due east, then one mile due north. you are now back at your house ! where do you live? there is more than one solution; find as many as possible.
There are infinitely many possible locations for your house that would allow you to travel one mile south, one mile east, and one mile north and end up back at your starting point.
As per the question given,
This is a classic puzzle known as the "One-Mile Puzzle". There are actually infinitely many possible locations for your house that satisfy these conditions.
One way to think about it is to consider the fact that the one-mile distances you traveled form the sides of a triangle. Since you traveled one mile due south, one mile due east, and one mile due north, the triangle you formed is an isosceles right triangle, with the right angle at the point where you ended up.
If you let (x, y) be the coordinates of your house, then the point where you ended up after traveling one mile south, one mile east, and one mile north is (x, y-1) + (1, y) + (x, y+1), which simplifies to (2x+1, 2y).
So, any point (x, y) that satisfies the equation 2x + 1 = 2y will work as a possible location for your house. For example:
If x = 0, then y = 1/2, so your house could be located at (0, 1/2).
If x = 1, then y = 3/2, so your house could be located at (1, 3/2).
If x = -1, then y = -1/2, so your house could be located at (-1, -1/2).
And so on...
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A rectangle flower garden is 9 2/5 feet long and 6 1/2 feet wide.What is the area of the flower garden?
Answer: 61 1/10
Step-by-step explanation:
9 2/5 x 6 1/2
I need help on this please
Question 2 (2 points)
About what percent of the data falls after 34?
20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
_________டப
25%
50%
75%
100%
From the given data points, about 50% of the data points are above the point 34.
What does a Percentage define?Percentage defines the parts of a number per fraction of 100 or a part per 100. It is usually denoted by the symbol '%'.
Percentage of a number is calculated by dividing the number with the whole and then multiply with 100.
Given are a set of data points.
20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
The numbers are in ascending order.
Total number of data points = 16
Data points above 34 are,
36 38 40 42 44 46 48 50
Number of data points which are above 34 = 8
Percent of numbers above 34 = (8 / 16) × 100 = 50%
Hence the percent of the data falls after 34 are 50%.
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hugh poured 100 kilograms of water into 10 kg of salt. He made ...... kilograms of ......% salt solution
Hugh poured 100 kilograms of water into 10 kg of salt. He made 110 kilograms of 9.09 % salt solution.
Finding the concentration of a solution:To find the mass of the solution add the mass of salt and the mass of the solution. To find the concentration of the solution divide the salt mass by the total mass of the solution and multiply by 100.
Here we have
Hugh poured 100 kilograms of water into 10 kg of salt.
After adding 100 kilograms of water to 10 kg of salt.
The total mass of the solution = 100 + 10 = 110 kg
The concentration of salt
= [ Mass of salt/ Mass of solution] × 100
= [ 10/110] × 100 = 9.09%
Therefore,
Hugh poured 100 kilograms of water into 10 kg of salt. He made 110 kilograms of 9.09 % salt solution.
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Determine the lengths of the sides of the rectangle using the given area. Give answers both exactly and approximately (to the nearest tenth). The area of the rectangle is 45cm-. The width of the rectangle is cm. The length of the rectangle is. cm.
Answer: Let the width of the rectangle be w and the length be l. We are given that the area of the rectangle is 45 cm^2, so we can write the equation:
w * l = 45
To find the lengths of the sides, we need to solve for w and l. We can divide both sides of the equation by any non-zero number to find an equivalent expression. For example, we can divide both sides by 9:
(w/9) * (l/9) = 45/9
Simplifying, we get:
w/9 * l/9 = 5
So, w/9 = 5/l and w = 5l/9.
Substituting the expression for w into the original equation, we get:
(5l/9) * l = 45
Expanding and simplifying, we get:
5l^2/9 = 45
Multiplying both sides by 9, we get:
5l^2 = 405
Dividing both sides by 5, we get:
l^2 = 81
Taking the square root of both sides, we get:
l = 9
So, the length of the rectangle is 9 cm. To find the width, we can substitute the value of l back into the expression we derived earlier:
w = 5l/9
w = 5 * 9 / 9
w = 5
So, the width of the rectangle is 5 cm.
The lengths of the sides of the rectangle, both exactly and approximately, are 9 cm and 5 cm.
Step-by-step explanation:
Find the value of each trigonometric ratio. Express your answer as a fraction in lowest terms.
The value of the trigonometric ratios are
1. sin C = 21/29
2. sin C = 3/5
3. cos C = 12/13
4. cos C = 8/17
5. tan A = 12/35
6. tan X = 4/3
How to find the trigonometric ratiosThe trigonometry problem in a triangle is worked using an acronym SOH CAH TOA, this acronym means:
Sin = opposite / hypotenuse - SOH
Cos = adjacent / hypotenuse - CAH
Tan = opposite / adjacent - TOA
1. sin C
sin C = opposite / hypotenuse = 21/29
3. cos C = 12/13
cos C = adjacent / hypotenuse = 36/39 = 12/13 to the lowest term
6. tan X = 4/3
tan X = opposite / adjacent = 36/27 = 4/3 to the lowest term
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T and W are the midpoints of the legs,
SU
and
VX
, of trapezoid SUVX.
If TW=30 and UV=40, what is SX?
Answer:
SX = 20
Step-by-step explanation:
the midsegment of a trapezoid is half the sum of the parallel bases, then
[tex]\frac{UV+SX}{2}[/tex] = TW , that is
[tex]\frac{40+SX}{2}[/tex] = 30 ( multiply both sides by 2 to clear the fraction )
40 + SX = 60 ( subtract 40 from both sides )
SX = 20
910x+45=28.3
can you please help me i have been on this question forever
Answer:
x = -0.01835
Step-by-step explanation:
1) Isolate the term with the variable, 910x, by subtracting 45 on both sides:
910x + 45 = 28.3910x + 45 - 45 = 28.3 - 45910x = -16.72) Divide both sides by 910 to solve for x
i need help with this pls
Answer:
90°
Step-by-step explanation:
every inscribed triangle with the baseline (Hypotenuse) being a diameter of the circle is a right-angled triangle.
therefore, that angle x = 90°.
