Prove that the angle between any two diagonals of a cube is given by cos=1/3
Consider a cube of side a units, with sides along x, y and z axis and one vertex at origin, as in the figure.
Here AB and CD are two diagonals of the cube, because A and B are diagonally opposite corners, also C and D.
In the figure, coordinates of A is (0, 0, 0) and that of B is (a, a, a). Then vector AB is given by,
[tex]\vec{\rm{AB}}=\rm{\left < a,\ a,\ a\right > }[/tex]
Coordinates of C is (a, 0, 0) and that of D is (0, a, a). Then vector CD is given by,
[tex]\vec{\rm{CD}}=\rm{\left < -a,\ a,\ a\right > }[/tex]
The diagonals each have a magnitude of a√3.
[tex]|\vec{\rm{AB}}|=|\vec{\rm{CD}}|=\rm{a\sqrt3}[/tex]
In the figure θ is the angle between the diagonals.
We take the dot product of the vectors AB and CD to get angle between them.
[tex]\longrightarrow\vec{\rm{AB}}\cdot\vec{\rm{CD}}=|\vec{\rm{AB}}|\cdot|\vec{\rm{CD}}|\cdot\cos\theta[/tex]
[tex]\longrightarrow\rm{\left < a,\ a,\ a\right > \cdot\left < -a,\ a,\ a\right > =a\sqrt3\cdot a\sqrt3\cdot \cos\theta}[/tex]
[tex]\longrightarrow\rm{-a^2+a^2+a^2=3a^2\cos\theta}[/tex]
[tex]\longrightarrow\rm{a^2=3a^2\cos\theta}[/tex]
[tex]\longrightarrow\rm{\cos\theta=\dfrac{a^2}{3a^2}}[/tex]
[tex]\longrightarrow\rm{\underline{\underline{\cos\theta=\dfrac{1}{3}}}}[/tex]
The angle between the diagonals satisfy this equation.
Hence Proved!
lee has $\frac{5}{7}$ of a gallon of water. he gives $\frac{1}{2}$ of a gallon of water to his dog, gilligan. what fraction of the amount of water that lee had originally did he give to gilligan?
The fraction Lee give the dog Gilligan is 7/10
How to determine the fraction given to the dogFrom the question, we have the following parameters that can be used in our computation:
Initial amount of water = 5/7 gallons
Amount of water given to the dog = 1/2 gallons
The fraction given to the dog is then calculated as
Fraction given to the dog = Amount of water given to the dog/Initial amount of water
Substitute the known values in the above equation, so, we have the following representation
Fraction given to the dog = (1/2)/(5/7)
Express as products
Fraction given to the dog = 1/2* 7/5
Evaluate the products
Fraction given to the dog = 7/10
Hence, the fraction is 7/10
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find an antiderivative of f(x) = tan−1x satisfying f(0) = 1.
The antiderivative of f(x) = tan⁻¹(x) is x tan⁻¹(x) − ln(x²+1)/2 + C
The term antiderivative in math is referred as a function that reverses what the derivative does
Here we have given the expression f(x) = tan⁻¹(x).
Now, the antiderivative of the expression is calculated by doing the integral
=> ∫tan⁻¹(x)dx
By using the integration by parts ∫udv = uv−∫vdu
Here let us consider that u=tan⁻¹(x) and dv=dx
And the value of du is calculated as,
=> du = ∫tan⁻¹(x)dx = dx/(x² + 1)
And the value of v = x.
Then the expression is written as,
=> ∫tan⁻¹(x)dx = x tan⁻¹(x) - ∫x/(x² + 1)dx
As per the concept of the constant of integration, it can be simplified as,
=> x tan⁻¹(x) − ln(x²+1)/2 + C
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what is meant by the statement that two variables are related? choose the correct answer. two variables are related when a scatterplot can be made of points using the two variables as values. two variables are related when a change in one can be shown to cause a change in the other. two variables are related when a discernible pattern exists between them. two variables are related when the value of one can be derived mathematically from the value of the other.
The statement that correctly describes related variables is given as follows:
Two variables are related when a discernible pattern exists between them.
What are related variables?Two variables are said to be related if one variable can be written as a function of another, that is, the output variable can be written as a function of the input variable.
There are multiple relations between variables, such as linear, quadratic and exponential relations. What is common for all these relations is that a pattern exists between these variables, meaning that the third option gives the correct option.
The pattern is identified from a sample of observations of these two variables, in which it is verified how the dependent variable varies according to the independent variable.
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Which situation best represents causation?
A. As the weight of a dog increases, the length of its tail decreases.
B. As the number of miles driven increases, the amount of gas used increases.
C. As the number of siblings a student has increases, the grade in the student’s class increases.
D. As the sale of winter coats increases, the amount of hot chocolate sold decreases.
The best represents causation is D, As the sale of winter coats increase the amount of hot chocolate sold decreases.
What is causation?Causation can be defined as the capacity of one variable to influence another. The first variable may bring the second into existence.
There are two types of causation:
factual cause is often established using the but for testProximate causation refers to a cause that is legally sufficient to find the defendant liableTherefore the correct option is D because the best way to represent causation is when a sale of winter coat increases the amount of hot chocolate sold decreases.
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What is the rational root theorem formula?
The rational zero theorem states that each rational zero(s) of a polynomial with integer coefficients f(x) = [tex]a_{n} x^{n} +a_{n-1} x^{n-1} +...........+a_{1} x^{} +a_{0}[/tex]
is of the form p/q, where:
->p is a factor of the constant
->q is a factor of the leading coefficient
->p and q are relatively prime (i.e., the fraction p/q is in its simplest form)
The rational root theorem, also known as the rational root test, is an algebraic theorem that states that for a polynomial equation in one variable with integer coefficients to have a solution (root) that is a rational number, both the constant term (the term without a variable) and the leading coefficient (the coefficient of the highest power) must be divisible by the fraction's denominator.
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Given that a = 6i- 3j, find:
a the unit vector in the same direction of a
b the angle that a makes with the positive x-axis.
The unit vector in the same direction will be equal to (6 i/5.1 - 3 j/5.1).
What is a Vector?A vector is a quantity with both magnitude and direction in physics. It is often depicted by an arrow with the same direction as the amount and a length proportionate to the size of the quantity.
A vector lacks position, but has magnitude and direction. A vector is therefore unaffected by displacement parallel to itself as far as its length is unaltered.
As per the given data in the question,
The unit vector is calculated as:
[tex]\vec u_a=\frac{\vec a}{||\vec a||}[/tex]
[tex]\vec a= x \vec i+y \vec j[/tex]
Now, calculate the magnitude of the vector,
[tex]\vec a[/tex] = √(x² + y²)
[tex]|\vec a|[/tex] = √6² - 3²
= √36 - 9
= 5.1
So, the unit vector will be,
[tex]\vec u_a[/tex] = (6 i - 3 j)/5.1
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Robert record the number of page he read each day. 27,54,81,108. What i the total number of page Robert will have read after 8 day?
The number of pages Robert reads each day form an arithmetic sequence 27, 54, 81, 108,... After 8 days he has read 216 pages.
The n-th term of an arithmetic sequence is given by:
a(n) = a0 + (n - 1) d
Where:
a0 = first term
d = common difference
In the given problem, the number of pages form an arithmetic sequence:
27, 54, 81, 108, ....
with:
first term = a0 = 27
common difference = d = 54 - 27 = 27
Hence, for n = 8
a(8) = 27 + (8 - 1) x 27
= 27 + 7 x 27 = 216
Hence, after 8 days he has read 216 pages.
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How do you know which graph is greater?
To determine which graph is greater, you need to compare the y-values of the two graphs at the same x-values or look at the overall shape of the graphs.
You can compare the y-values of the two graphs at the same x-values to determine which graph is greater if one graph has a higher y-value than the other graph at a particular x-value, then the first graph is greater at that x-value.
For example, suppose you have two graphs: one is a straight line with a slope of 2 and the other is a straight line with a slope of -3. If you compare the two graphs at the same x-values, you will see that the graph with the slope of 2 has a higher y-value than the graph with the slope of -3 for all x-values. This means that the graph with the slope of 2 is always greater than the graph with the slope of -3.
You can also compare the graphs by looking at the overall shape of the graphs. For example, if one graph is always above the other graph, then it is clear that the first graph is greater. On the other hand, if one graph crosses over the other graph, then there may be some x-values for which one graph is greater and some x-values for which the other graph is greater.
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What are the zeros of the function x2 5x 5?
The zeros of the function x^2 + 5x + 5 = 0 are (-5 ± sqrt (5))/2.
We know that the general form of a quadratic equation is:
ax^2 + bx + c =0
The equation we have is:
x^2 + 5x + 5 = 0
Comparing the given equation with the general form of a quadratic equation, we get that:
a = 1, b= 5, and c = 5
To find the zeros of the any quadratic function, we use the formula:
x = (-b ± sqrt (b^2 - 4ac))/(2a)
Therefore, the zeros of the equation x^2 + 5x + 5 = 0 are:
x = (-5 ± sqrt (5^2 - 4x(1)x(5))/(2x1) = (-5 ± sqrt (5))/2.
The question is incomplete, the complete question is:
What are the zeros of the function x^2 + 5x + 5 = 0?
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What additional information do you need to conclude that the two triangle given here under are congruent using SAS rule?
According to the SAS rule, the two triangles are congruent.
In geometry SAS rules states that if two triangle are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle.
Here we have to prove that the two triangle given here under are congruent using SAS rule.
While we looking into the following diagram, we have identified that the following are given,
In the given triangle, we have know that that the triangle △GHJ and △RTS, has the side values HJ =TS
Here we also know that the angles ∠GHJ=∠STR are equal.
Then the sides of the triangle HG = TR
Therefore, the △GHJ Congruent to △RTS
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see picture below!!!!!!
will mark brainlest pls help me
Answer:
c. 49
Step-by-step explanation:
(g o f) (3)
f(x) = 3x - 2
g(x) = x²
(g o f) (x) = (3x - 2)²
= (3 × 3 - 2)²
= (9 - 2)²
=7²
=49
a triangle has sides of 12,36,37 is this a right triangle
The triangle has sides of 12, 36, and 37 and is not a right triangle.
What are Pythagoras's triples of the right triangle?
Pythagorean triples are defined as a² + b² = c², where a, b, and c are positive integers. These triples are denoted as (a,b,c). The perpendicular is a the base is b, and the hypotenuse is c of a right-angled triangle.
We have,
a triangle has sides of 12, 36, 37
A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.
(12)² + (36)² = (37)²
1440 = 1369
S0, LHS ≠ RHS
Hence, the triangle has sides of 12, 36, and 37 is not a right triangle.
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HELP ASAP PLS!!!!!!!!!!!!!!
Answer:
x=5
Step-by-step explanation:
supplementary angles adds to 180
25x+5+10x=180
35x+5=180
35x=180-5
35x=175
divide
x=5
, if "y varies directly as x", find the constant of variation and write an equation of direct variation that
relates the two variables.
y = 0.4 when x = -1
Answer:
k = - 0.4 , y = - 0.4x
Step-by-step explanation:
given y varies directly as x then the equation relating them is
y = kx ← k is the constant of variation
to find k use the condition y = 0.4 when x = - 1 , then
0.4 = - k ( multiply both sides by - 1 )
- 0.4 = k
then constant of variation is k = - 0.4
y = - 0.4x ← equation of variation
How do you tell the difference between a hole and an asymptote?
As per the concept of the function, the difference between a hole and an asymptote is explained below
The term function in math is known as the special relationship where each input has a single output.
Here we need to find the the difference between a hole and an asymptote.
In order to find the difference, here we must know the definition is asymptote.
The term asymptote is defined as a straight line that constantly approaches a given curve but does not meet at any infinite distance.
Based on the definition of asymptote we have identified the difference between these are that the holes occur when factors from the numerator and the denominator cancel. Where as when a factor in the denominator does not cancel, it produces a vertical asymptote.
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at a sporting event, cheerleaders will throw 50 bundled t-shirts into the crowd. the t-shirt sizes consist of 10 small, 15 medium, and the remainder either large or extra large. suppose ana catches a t-shirt. what is the probability that she will catch a t-shirt that is not a size small? responses 0.10 0.10 0.20
There is a 4/5 probability of catching a T-shirt that is not a size small.
What is probability?Simply put, the probability is the likelihood that something will occur.
When we don't know how an event will turn out, we can discuss the likelihood or likelihood of several outcomes.
Statistics is the study of events that follow a probability distribution.
So, there were 50 shirts thrown into the throng in total.
There were 10 little shirts thrown.
15 medium-sized shirts were hurled.
The quantity of extra-large and big shirts.
= Total shirts - ( 10 + 15)
= 50 - 25 = 25
So, a total of 25 shirts are scattered throughout the large/extra large shirts.
A number of shirts that are NOT SMALL size are now available.
= Total shirts - Shirts with small size = 50 - 10 = 40
Probability of getting a not small t-shirt = The total not small t-shirt/The total number of t-shirts
= 40/50
= 4/5
Therefore, there is a 4/5 probability of catching a T-shirt that is not a size small.
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The table below contains a and b in equivalent ratios. Fill in the missing value in the table.
A
24
B
15
C
20
Answer: 24
Step-by-step explanation:
1*4=4
2*4=8
3*4=12
6*4=24
9*4=36
What is the relationship of the volume of the pyramid to the volume of prism the volume of a pyramid is blank the volume of a prism with the same base area and height?
The relationship of the volume of the pyramid to the volume of prism: The volume of a pyramid is 1/3 the volume of a prism with the same base area and height.
In this question we need to find the relationship of the volume of the pyramid to the volume of prism.
The volume of a pyramid is ________ the volume of a prism with the same base area and height.
We know that the formula for the volume of pyramid.
V1 = 1/3 * base area * height
And the formula of the volume of prism is:
V2 = base area * height
We have been given a pyramid and prism have the same base area and height.
So, volume of pyramid = 1/3 * volume of prism
Consider a ratio the volume of the pyramid to the volume of prism.
(the volume of the pyramid) / (the volume of prism) = 1/3
Therefore, the volume of a pyramid is 1/3 the volume of a prism with the same base area and height.
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What is the slope of the line represented by the equation y 1 3x 4?
1/3 is the slope of the line represented by the equation y = 1/3x + 4.
Define slope.In other words, the slope is the ratio of the change in height to the horizontal distance between any two locations on the line. It is calculated as the ratio of the "rise" divided by the "run" between two points on a line. The angle that a straight line forms with the positive x-axis direction is known as its inclination. The slope or gradient of a line is the tangent of the angle it creates with the positive x-axis direction. A line's slope is typically represented by the symbol m.
Given
Equation
y = 1/3x + 4
Using slope intercept form,
y = mx + b
m = slope
m = 1/3
1/3 is the slope of the line represented by the equation y = 1/3x + 4.
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How do you find the altitude of a triangle on a calculator?
The altitude of the triangle is determined by, a = √(c² - b²)
Given,
Altitude of triangle;
A triangle's altitude is determined by drawing a perpendicular line from its vertex to its opposite side. The altitude, sometimes referred to as the triangle's height, forms a right-angle triangle with the base.
Here,
The simplest way is to use the Pythagorean theorem if you are aware of two additional right triangle sides:
a² + b² = c²
Where,
a is the altitude, b is the base and c is the hypotenuse
Then,
If leg a is the missing side, change the equation such that a is on one side and compute the square root of that number:
a = √(c² - b²)
That is,
We can find the altitude of the triangle by, a = √(c² - b²)
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What is an example of an inverse function?
Suppose that f and g are inverse functions, with f(g(x)) = g(f(x)) = x. A function's inverse retrieves the initial value, and g(y) = (y-5)/2 = x is the inverse of f. (x).
What is a inverse function theorem?The inverse function theorem in mathematics provides a necessary requirement for a function to be invertible in the vicinity of a point in its domain, meaning that its derivative is continuous and non-zero at the point. This condition is relevant to differential calculus.
The derivative of the inverse function is also given a formula by the theorem. This theorem in multivariable calculus can be extended to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, providing a formula for the Jacobian matrix of the inverse.
The inverse function theorem also applies to complex holomorphic functions, differentiable functions between Banach spaces, differentiable functions between manifolds, and so forth.
Picard and Goursat used an algebraic approach to prove the theorem in the beginning.
Hence, Suppose that f and g are inverse functions, with f(g(x)) = g(f(x)) = x. A function's inverse retrieves the initial value, and g(y) = (y-5)/2 = x is the inverse of f. (x).
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Find: P(Small and Blue)
Remember to reduce your answer.
Large | Small
Red | 17 | 3
Blue | 8 | 12
HURRY!!
Red to tiny will have a probability of 12/20.
Describe probability.
In other terms, probability is a number that indicates the likelihood of an event occurring. Probability is defined as the ratio of favorable outcomes to all outcomes.
Probability is calculated as follows: N = N samples / N favorable outcomes
There are 20 blue numbers, of which the large ones are 17 and the small ones 3. Likewise, there are 20 red digits; the large ones are 8 and the small ones are 12.
The likelihood of receiving a little red number will be determined as follows:
Probability is calculated as follows: N = N samples / N favorable outcomes
12 good outcomes total.
There are 20 samples total.
Fill out the formula above with the value.
12/20 is the probability.
As a result, 12/20 will be the likelihood of red to little.
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What is f 0 in a function?
The expression f (0) in a function is used to represents the y-intercept on the graph of the function f (x).
The point where the graph cuts the y - axis is known as the y - intercept whereas the point where it cuts the x - axis is known as x - axis. The graph cuts y- axis when the value of x coordinate is 0.
The values which satisfies the equation is known as the zeros of the function. The graph of a function is the set of all points in the plane of the form (x f(x)).
The graph of f could similarly be defined as the graph of the equation y = f(x). As a result, the graph of a function is a subset of the graph of an equation.
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Rearrange the formula and solve for r.
V = 1/3πr²h
•Mutiply both sides by 3 to cancel out the fraction
3V = πr²h
•Divide both sides by πh 3V/πh = r²
•Find the square root of both sides
√3V/πh = r
•Switch the sides to make it easier to read
r = √3V/πh
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literally anybody help me please!! all you have to do is find the slope and y intercept for the equation on the picture. PLEASE HELP ME ASAP!!
The y-intercept is the point on the graph where it crosses the y-axis, and has coordinates of (0, b). The y-coordinate is the value of b in the slope-intercept form. The Slope is - ⅖.
What is the Slope and Y-intercept of a Line?Y-intercept = (0, 4), where b = 4
Given the linear equation in standard form, 2x + 5y = 20, where A = 2, B = 5, and C = 20:
Start by transforming the standard equation into its slope-intercept form, y = mx + b, where m = slope, and b = y-intercept.
Subtract 2x from both sides:
2x -2x + 5y = - 2x + 20
5y = -2x + 20
Divide both sides by 5 to isolate y:
y = - ⅖x + 4 ⇒ This is the slope-intercept form where the slope, m = -⅖, and the y-intercept, b = 4. The y-intercept is the point on the graph where it crosses the y-axis, and has coordinates of (0, b). The y-coordinate is the value of b in the slope-intercept form. Therefore, the y-intercept is (0, 4).
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help me out on this please
We need to use the concept of Finding zeros of the polynomial function. The zeros of the function are 1/2,1+2i,1-2i.
How do you find the zeros of the polynomial?The values of x that fulfill the formula f(x) = 0 are the zeros of a polynomial. The polynomial's zeros are the x values for which the function's value, f(x), equals zero in this case. The degree of equation f(x) = 0 determines how many zeros a polynomial has. The zeros of the polynomial are all such domain values of the function for which the range is zero.
The locations where the graph of y = f(x) crosses the x-axis are known as the polynomial's zeros graphically. In the information that follows, we will learn more about how to express a polynomial's zeros on a graph.
Calculation:Take f(x)=2x^3 -5x^2+12x-5
So in the question he mentioned that (2x-1) is factor of the polynomial.
So take out (2x-1) from the polynomial function
f(x)=(2x-1)(x^2-2x+5)
We know that zeros of a polynomial can be found when f(x)=0
⇒(2x-1)=0
∴x=1/2
⇒(x^2-2x+5)=0
By using the concept of finding roots of a quadratic equation
(-b±√b2-4ac)/2a
⇒(2±4i)/2
∴x=(1+2i) and (1-2i)
∴1/2,(1+2i),(1-2i) are the zeros.
We need to use the concept of Finding zeros of the polynomial function. The zeros of the function are 1/2,1+2i,1-2i.
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What is the 3x 1 problem called?
3x + 1 is called the Collatz problem.
Describe Collatz problem."The 3x+1 problem, often referred to as the Syracuse problem, Collatz problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns the behaviour of the iterates of the function that takes odd numbers n to 3n+1 and even integers n to n/2. One of the most well-known unsolved mathematical puzzles is the Collatz conjecture. The conjecture asks if every positive integer will eventually become 1 if two basic arithmetic processes are repeated.
Given
3x + 1 is called the Collatz problem.
"The 3x+1 problem, often referred to as the Syracuse problem, Collatz problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns the behaviour of the iterates of the function that takes odd numbers n to 3n+1 and even integers n to n/2.
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What is a multiples of 16?
several sixteens. Numerous numbers that are multiples of 16 include 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, and 176.
What is the trick to finding multiples of a number?In mathematics, a multiple is any number that results from multiplying a given number by an integer. As an illustration, multiples of 5 include: 10, 15, 20, 25, 30, etc. 14, 21, 28, 35, 42, 49, and so on are all multiples of 7.
A number is a multiple of a number if another natural number and the given number are multiplied together.
Divide the digits into pairs starting at the end, then multiply the final group by 1, the group before it by 2, the group before that by 4, then 8, 16, and so on. Include every component. If the outcome is a multiple of 7, then so was the original number. Since 91 (137) is a multiple of 7, so is 21553.
16-digit multiples. There are 16 multiples, including 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, and 176.
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What is the measure of
Angles can be measured using one of two standard units. Degrees are the most widely used unit of measurement.
∠ABD = 39°
What is meant by angle?An angle is a figure in Euclidean geometry made up of two rays that share a common terminal and are referred to as the angle's sides and vertices, respectively. Two rays can form angles in the plane where they are positioned. Angles are also produced when two planes intersect. These are what are known as dihedral angles.An angle is a figure in plane geometry that is created by two rays or lines that have a shared endpoint. The Latin word "angulus," which meaning "corner," is the source of the English term "angle." The shared terminus of two rays is known as the vertex, and the two rays are referred to as sides of an angle.Therefore, the correct option is c) 39°
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