The direction cosines of a vector are the cosines of the angles between the vector and the three mutually perpendicular axes of a coordinate system.
The direction cosines of a vector [tex]$\mathbf{a}$[/tex] are denoted by l, m and n. Mathematically, the relation between these direction cosines is given by:
[tex]$$\mathbf{a}=l\mathbf{\hat{i}}+m\mathbf{\hat{j}}+n\mathbf{\hat{k}}$$[/tex]
where i,j and k are unit vectors along the three axes of the coordinate system.
To calculate the direction cosines of a given vector [tex]$\mathbf{a}$[/tex] with components [tex]$a_x$, $a_y$ and $a_z$[/tex], we can use the following formula:
[tex]$$l = \frac{a_x}{\sqrt{a_x^2+a_y^2+a_z^2}}$$[/tex]
[tex]$$m = \frac{a_y}{\sqrt{a_x^2+a_y^2+a_z^2}}$$ \\ $$n = \frac{a_z}{\sqrt{a_x^2+a_y^2+a_z^2}}$$[/tex]
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What is the perimeter of a 30 60 90 triangle whose shorter leg is 5 inches long?
The perimeter of the triangle is 23.66 inches
According to the 30-60-90 triangle theorem, the hypotenuse is twice as long as the shorter leg. Additionally, the sides of this particular triangle have a ratio of 1:2: [tex]\sqrt{3}[/tex].
Now in the question
As shown in the diagram, if the shorter leg of the triangle at 30°, 60°, and 90° is extended in the direction of the right angle by an amount equal to itself and connected to the third point, we obtain a triangle that is congruent with the original triangle. It is clear that the combined triangle is an equilateral triangle with equal sides and angles.
The greatest leg, AB, in the picture is therefore twice as long as the shortest leg of the original triangle and equal to base BC.
Therefore according to the question
given that the shorter leg is 5 inches,
hypotenuse = 5*2 = 10 inches.
the longest leg = 5[tex]\sqrt{3}[/tex], and
The perimeter is thus 5 + 10 + 5[tex]\sqrt{3}[/tex] = 15 + 5[tex]\sqrt{3}[/tex] = 23.66 inches
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in triangle ABC,AC=BC,CD is perpendicular to AB with D belonging to AB,AB=4 in. and CD=\sqrt(3) in.find AC
The length of the triangle ΔABC will be √7 inches.
What is a Pythagoras theorem?The Pythagoras theorem states that the sum of two squares equals the squared of the longest side.
The Pythagoras theorem formula is given as
H² = P² + B²
In triangle ΔABC, AC = BC, the CD is perpendicular to AB with D belonging to AB, AB = 4 inches, and CD = √(3) inches.
The CD is half of the length AB. Then we have
CD = AB / 2
CD = 4 / 2
CD = 2 inches
Then the length of AC is calculated as,
h² = (√3)² + (2)²
h² = 3 + 4
h= √7 inches
The length of the triangle ΔABC will be √7 inches.
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The circumference of a circle can be represented by the equation C= 2πr. Rewrite
this equation to represent the value of r in terms of C and π. Use your new equation to find the radius of a circle that has a circumference of 54 inches. Leave your answer in terms of pi and solve algebraically
Answer:
see explanation
Step-by-step explanation:
C = 2πr ( r is the radius )
isolate r by dividing both sides by 2π )
[tex]\frac{C}{2\pi }[/tex] = r
given C = 54 , then
r = [tex]\frac{C}{2\pi }[/tex] = [tex]\frac{54}{2\pi }[/tex] = [tex]\frac{27}{\pi }[/tex] inches
The ratio of the number of report cards Ms. Wickham folded to the number that Ms. Cho folded was 2:3. At the end of the day, Ms. Leiva and Mr. Gamory folded 300 report cards. Who folded more report cards and by how many
Ms. Wickham folded 200 report cards and Ms. Cho folded 300 report cards, giving Ms. Cho an average of 100 report cards.
Ms. Wickham and Ms. Cho together folded 500 report cards.
Ms. Wickham folded 200 report cards
(200/500 = 2/5)
Ms. Cho folded 300 report cards
(300/500 = 3/5)
Mr. Leiva and Mr. Gamory folded 300 report cards.
Ms. Cho folded more report cards than Ms. Wickham, by
100 (300 - 200 = 100).
Ms. Wickham and Ms. Cho together folded 500 report cards, with the ratio of Ms. Wickham's report cards to Ms. Cho's being 2:3. Ms. Wickham folded 200 report cards, while Ms. Cho folded 300. At the end of the day, Mr. Leiva and Mr. Gamory folded an average 300 report cards. This meant that Ms. Cho folded more report cards than Ms. Wickham, by 100. Therefore, Ms. Cho folded 400 report cards in total, while Ms. Wickham only folded 300.
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What are the x-intercept and the y intercept of the graph of 6x - 3y = 36?
The x-intercept and the y intercept of the graph of 6x - 3y = 36 are x = 6 and y = -12
How to determine the x-intercept and the y intercept of the graph?
From the question, we have the following parameters that can be used in our computation:
6x - 3y = 36
Set y to 0 to determine the x-intercept
So, we have the following representation
6x = 36
Divide by 6
x = 6
Set x to 0 to determine the y-intercept
-3y = 36
Divide by -3
y = -12
hence, the intercepts are x = 6 and y = -12
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The diameters of the top and bottom of a frustum of a cone are 6cm and
15cm respectively. If the height of the frustum is 30cm, calculate (i) the
curved surface area (ii) the volume of the frustum.
Curved surface area of frustum is 1000.82 cm² and
Volume of frustum is 2756.754 cm³.
What is a conical frustum?Frustum of cone is the part of cone when it is cut by a plane into two parts. The upper part of cone remains same in shape but the bottom part makes a frustum.
Frustum is a Latin word which means ‘piece cut off’. When a solid (generally a cone or a pyramid) is cut in such a manner that base of the solid and the plane cutting the solid are parallel to each other, part of solid which remains between the parallel cutting plane and the base is known as frustum of that solid.
Height of frustum = h = 30 cm
Diameter of top of frustum = 6 cm
Radius of top = R = 6/2 = 3cm
Diameter of bottom of frustum = 15 cm
Radius of bottom = r = 15/2 = 7.5 cm
Slant Height = l = √h²+ (r - R)²
⇒ l = √30² + (7.5 - 3)²
⇒ l = √900 + 20.25 = √920.25 = 30.34 cm
A) Curved surface area of frustum = π(r + R)l
= 3.1416 × (7.5+3) × 30.34
= 3.1416 × (10.5) × 30.34
Curved surface area of frustum = 1000.82 cm²
B) Volume of frustum = (πh/3)(r²+R²+r.R)
= (3.1416 × 30/3)(7.5² + 3² + 7.5×3)
= (31.416) × (56.25 + 9 + 22.5)
= 31.416 × 87.75
Volume of frustum = 2756.754 cm³
Hence, the curved surface area and volume of given conical frustum are 1000.82 cm² and 2756.754 cm³ respectively.
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carmen has 92 kilograms of clay. she has 5 students in her art class. how much clay does each student get
Answer:
18.4 kg of clay
Step-by-step explanation:
92 kg for 5 students
=> 92 : 5 = 18,4 kg for 1 student
Answer: 18.4 kg
Step-by-step explanation:
DATA :
total mass of clay = 92 kg
total students = 5
how much clay does each student get?
SOLUTION:
assuming that each student got equal amount of clay, we are simply going to divide ⇒92 by 5.
⇒92 ÷ 5
⇒18.4 kg ANSWER
hope that helps...
Write a rule to calculate 1% of any number
Need help With answer
Answer:
Step-by-step explanation:
from the graph we konw the ∠ABD=∠DBC, because ∠DBC=20, so ∠ABD=20
A storm that moved through several states at a speed of 60 miles per hour affected 219 miles of land. What did that make the storm
The longest tornado ever recorded spanned 219 miles of land as it swept through many states at a pace of 60 miles per hour.
What are the importance of vector calculus?A tornado could cause a natural disaster to wreak havoc in significant geographic areas.A violent column of air between a thunderstorm and the earth is what is known as a tornado.In severe tornadoes, winds exceeding 200 miles per hour (h) are not uncommon.In conclusion, the longest tornado ever recorded affected 219 miles of land as it swept through many states at a speed of 60 miles per hour.Mathematically, a tornado is vector calculus with a fury. Severe storms are the result of the interaction of the two scalar forces of pressure and humidity with the vector field known as wind.To learn more about tornado, refer to:
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What is tautology in math?
A statement which is known as tautology is a type of compound statement in whose result is always the truth value. We don't take in consideration the other individual values in consideration , the result in tautology is always true.
It is one of the most significant part in logical mathematics if we need to find the most accurate answers or findings among a given group of answers.
In mathematics, tautology is used so as to ensure that the results which is obtained are true and exact and not false . A tautology is a compound statement and it will always returns true value. The result in tautology is always true, regardless of what the constituent parts are.
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What is 4 sided called?
Answer:
quadrilateral?
Step-by-step explanation:
By some estimates, about 30% of all males have a particular defect related to their eyes. How would you assign random numbers to conduct a simulation based on this
statistic?
Answer:
related to their eyes. How would you assign random numbers to conduct a simulation based on this
How do you find the domain and range of a function x 4?
The domain of x=4 is (−∞,∞) and range is {y|y=4} .
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
(−∞,∞)
{x|x ∈ R}
y=4
is a straight line perpendicular to the y-axis, which means that the range is a set of one value {y|y =4}
Set-Builder Notation:{y|y =4}.
The range of values that we are permitted to enter into our function is known as the domain of a function. The x values for a function like f make up this set (x). A function's range is the collection of values it can take as input. After we enter an x value, the function outputs this sequence of values.
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Audrey walked 285 meters west and Billy walked 213 meters south from the same starting location. How far apart are Audrey and Billy?
Audrey and Billy are as far as 367 meters apart.
What is displacement of an object?The displacement of an object is its distance covered or moved in a specific direction. Displacement is an example of vector quantity.
A vector quantity is generally any type of quantity which can be expressed in magnitude and direction. Some examples include: weight, displacement, acceleration, force, etc.
Since Audrey walked 285 meters west and Billy walked 213 meters south from the same starting location, then we can determined the distance far apart by applying the Pythagorean theorem.
So that,
/hyp/^2 = /opp/^2 + /adj/^2
Let their distance apart be represented by s,
s^2 = 231^2 + 285^2
= 53561 + 81225
s = (134586)^1/2
s = 366.86
Audrey and Billy are as far as 367 meters apart.
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At the opening session of the United States Supreme Court, each justice shakes hands with all the others. There are nine justices that sit on the Supreme Court. How many handshakes do they make during the opening session
According to probability, total number of handshakes will be 36.
How many handshakes are possible?Handshakes equal n * (n - 1)/2. This is the case because a handshake between two people is not counted twice and each of the n people can shake hands with n - 1 people (they wouldn't shake their own hand). Any number of people can be calculated using this formula.
The first person will need to shake eight (8) people's hands. Only seven (7) hands will need to be shook by the second person because the first handshake will not be repeated. Due to the same factors as mentioned above, only six (6) people will require a handshake from the third person. This continues until the final person has no one left to shake hands with because everyone has already done so.
The formula to calculate how many handshakes were exchanged is n = 8 + 7 + 6 +... + 1 = 36.
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5x6+78-12
help please
Answer:
96
Step-by-step explanation:
5 x 6 = 30
78 - 12 = 66
5 x 6 + 78 - 12
= 30 + 66
= 96
Answer:96
Step-by-step explanation:
QUESTION 1 OF 10
If √62-9 =z, what is the value of ?
The required value of z is "i√6" to the given expression√(6/2 - 9) = z.
What is a complex number?A complex number is a combination of real values and imaginary values. It is denoted by z = a + ib, where a, and b are real numbers and i is an imaginary number.
The expression is given in the question, as follows:
√(6/2 - 9) = z
We have to determine the value of z to the given expression.
As per the question, we have
⇒ √(6/2 - 9) = z
⇒ √(3 - 9) = z
⇒ √(- 6) = z
⇒ √(6 × -1 ) = z
⇒ (√6 × √-1 ) = z [∵ imaginary number i = √-1 ]
⇒ (√6 × i ) = z
⇒ z = i√6
Therefore, the required value of z is "i√6" to the given expression.
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The marked price of a computer is Rs. 60000. If allowing a% discount on the marked price and adding a% VAT, the price of the computer will be Rs. 58986. Find the rate of VAT.
The problem can be solved using algebra. Let's say the discount rate is x, then the price after discount is:
60000 (1 - x/100)
The price after VAT is:
60000 (1 - x/100) (1 + y/100) = 58986
Where y is the rate of VAT.
Rearranging the equation we get:
(1 - x/100) (1 + y/100) = 58986/60000
Multiply both sides by 100 to get rid of the fractions:
100 (1 - x/100) (1 + y/100) = 981
Expanding and grouping like terms, we get:
100 - x + 100y - xy = 981
We can see that x is in two terms, so we can move one side of the equation and get:
100 - x + 100y = 981 + x
x = (981 + x - 100 + 100y)/100 = (981 + xy - 100y)/100
Now we have x in only one term and we know the value of x is 10%, so we can substitute it:
10 + 10y = 9.81
Solving for y we can find that y = 0.81 or 81%
So the VAT rate is 81%
The table shows a bacterial population y after x days
days, X 0,1,2,3,4
Population,8,24,72,216,648
Write an exponential function that represents the population
Y=
The exponential function that represents the population is y = a*3^x.
What is exponential function?
The formula for exponential functions is y = ab^x, where b is the constant ratio and a is the starting point. Observe that as the value of x increases by a constant amount, the value of y increases by a common ratio by looking at a table of ordered pairs. All exponential functions share this trait.
We have to find the exponential function for the given table.
Since,
Exponential function = ab^x
where,
a is the starting point
b is the common ratio.
Here,
Common ratio = 24/8 = 72/24 = 216/72 = 648/216 = 3
So, the function becomes,
Exponential function = y = a*3^x
Hence, the exponential function that represents the population is
y = a*3^x.
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Write the Riemann sum to find the area under the graph of the function f(x) = x4 from x = 5 to x = 7.
Answer:
Refer to the attached image.
Step-by-step explanation:
I NEED HELP ASAP!!!!!!!!
Answer:
(- 1, 9 )
Step-by-step explanation:
6x - y = - 15 → (1)
x + 2y = 17 → (2)
multiplying (1) by 2 and adding to (2) will eliminate y
12x - 2y = - 30 → (3)
add (2) and (3) term by term to eliminate y
13x + 0 = - 13
13x = - 13 ( divide both sides by 13 )
x = - 1
substitute x = - 1 into either of the 2 equations and solve for y
substituting into (2)
- 1 + 2y = 17 ( add 1 to both sides )
2y = 18 ( divide both sides by 2 )
y = 9
solution is (- 1, 9 )
You have a 5 inch by 7 inch picture that you framed. When you hung the picture, you found that the entire area is 69.56 inches squared. What is the width of the frame?
The width of the frame is 2.4 inches
How to determine the width of the frame?
Let w represent the width of the frame
Then the area of the frame is:
(w + 5)(w + 7) = 69.56
Rewrite it in Standard form of a quadratic equation:
w² + 12w + 35 = 69.56
w² + 12w - 34.56 = 0
Use Quadratic Formula to solve for w:
w = (-b±√(b²-4ac))/(2a)
w = (-12±√(12²-4×1×(- 34.56)))/(2×1)
w = (-12±√(144+138.24))/(2)
w = (-12±16.8)/2
w = -6 ± 8.4
w = -6 + 8.4 or -6-8.4
w = 2.4 or w = -14.4
Since the width cannot be negative.
Thus, w = 2.4 inches
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A wedding menu has 5 starters, 3 mains, and 4 desserts. The list gives the meal options each attendee must choose from. a starter and a main, a main and a dessert, a starter, a main, and a dessert. How many different ways of choosing a meal for this wedding are there
There are 60 different ways of choosing a meal for this wedding.
How many different ways of choosing a meal?There were five distinct main courses available for selection.
There are a total of 15 possible main dish and salad combinations when ordering these 5 distinct main meals together with any one of the 3 available salads.
A total of 60 distinct main dishes, salads, and dessert combinations are conceivable by ordering one of the four potential deserts for each of the 15 various main meals and salad combinations.
60 distinct main dishes might be purchased since.
There are 60 different ways of choosing a meal for this wedding
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A recipe calls for 2 2/3 cups of flour to make two regular batches of cookies. Shiloh needs to make multiple batches of cookies. Represent this situation with an equation. What is the constant of proportionality? How can you identify it in an equation? How can you use an equation to determine if a relationship is proportional?
This situation can be represented with an equation as y = mx + b. The constant of proportionality (m) is a number that represents how much flour is needed for each additional batch of cookies. If the equation is in the form y = mx + b, then the relationship is proportional.
Shiloh needs to make multiple batches of cookies from a recipe that calls for 2 2/3 cups of flour to make two regular batches. To represent this situation in an equation, the equation is y = mx + b, where y is the amount of flour needed for the number of batches (x) Shiloh is making, m is the constant of proportionality, and b is the initial amount of flour needed (2 2/3 cups).
The constant of proportionality (m) in the situation above is the measure of how much flour is needed for each additional batch of cookies. For example, if the constant of proportionality is 0.75, that means Shiloh would need to add 0.75 cups of flour for each additional batch of cookies.
Therefore, if Shiloh is making three batches of cookies, the equation would be y = 0.75x + 2 2/3, and the total amount of flour needed would be 4 1/8 cups (2 2/3 + 0.75 + 0.75).
By using an equation to represent this situation, it is easy to determine if a relationship is proportional.
If the equation is in the form y = mx + b, then the relationship is proportional.
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PLEASE HELP FAST I WILL GIVE BRAINLIEST, Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.
Using the properties of integer exponents, match each expression with its equivalent expression.
The 5⁻³ is equivalent to 1/125, - 5⁻³ is equivalent to -1/125, (-5⁻³)⁻¹ is equivalent to -125 and (-5⁻³)⁰ is equivalent to 1
What is Expression?An expression is combination of variables, numbers and operators.
Equivalent expressions are expressions that work the same even though they look different.
5⁻³
This can be written as 1/5³ which is equal to 1/125
-5⁻³
This can be written as -1/5³ which is equal to -1/125
(-5⁻³)⁻¹
We have an exponential property
[tex](x^{a} )^{b} =x^{ab}[/tex]
So (-5⁻³)⁻¹=(-5)³=-125
(-5⁻³)⁻¹ equivalent expression is -125
(-5⁻³)⁰
=(-5)⁰=1
Hence, the 5⁻³ is equivalent to 1/125, - 5⁻³ is equivalent to -1/125, (-5⁻³)⁻¹ is equivalent to -125 and (-5⁻³)⁰ is equivalent to 1
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The sum of the digits of a two-digit number is 7. When the digits are reversed, the number increases by 45. What is the original number?
original number: ⬜
The original number is 25.
We know that,
The sum of the original number is 7 --(i)
When reversed, the value of the number is increased by 45 --(ii)
Therefore, adding (i) and (ii), we get:
7 + 45 = 52
Also, 5 + 2 = 7
Since 52 is the reversed number,
Therefore, the original number is 25.
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please help: find m∠BCD
Answer:
m<BCD=60
Step-by-step explanation:
4x+6=7x-12
3x=18
x=6
4(6)+6
24+6
30
30+30=60
======================================================
Reason:
Use the HL (hypotenuse leg) theorem to prove that triangle BCF is congruent to triangle DCF.
It then leads to the fact that angle BCF is congruent to angle DCF.
Set those angle expressions equal to one another. Solve for x.
angle BCF = angle DCF
4x+6 = 7x-12
4x-7x = -12-6
-3x = -18
x = -18/(-3)
x = 6
Then,
angle BCF = 4x+6 = 4*6+6 = 30angle DCF = 7x-12 = 7*6-12 = 30Both angles are the same measure to confirm the correct x value.
Lastly, add up those angles to get angle BCD.
30+30 = 60 degrees is the final answer.
The amount of eroded rock is measured in cubic kilometers (km3). A cubic kilometer is a cube that measures 1 km on each side. How many km3 of rock were eroded
The amount of eroded rock in cubic kilometres is 1 km3.
To calculate the amount of eroded rock in cubic kilometres, we need to find the volume of the area that has been eroded. The volume of a rectangular shape is found by multiplying the length, width, and height (or depth) together.
In this case, the area that has been eroded is 10 km long, 5 km wide, and 20 m deep. However, we need to convert the depth from meters to kilometres to match the units of the other measurements. Since 1 km = 1000 m, we can convert the depth by dividing by 1000:
20 m / 1000 = 0.02 km
Now that all of the measurements are in kilometres, we can find the volume by multiplying them together:
10 km * 5 km * 0.02 km = 1 km3
So the amount of eroded rock in cubic kilometres is 1 km3.
Complete Question:
The amount of eroded rock is measured in cubic kilometres (km3). A cubic kilometre is a cube that measures 1 km on each side. How many km3 of rock were eroded in an area that is 10 km long, 5 km wide, and 20 m deep?
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A sequence is represented by the explicit formula
The value of a₂₆ in the arithmetic sequence aₙ= 14 + 9n is 248.
The correct option is D.
What is an arithmetic sequence?A series of integers called an arithmetic progression or arithmetic sequence (AP) has a constant difference between the terms.
Given:
An arithmetic sequence,
aₙ= 14 + 9n
We have to find the value of a₂₆.
That means, n = 26.
Substitute the value of n = 26 to the sequence,
a₂₆ = 14 + 9(26)
a₂₆ = 14 + 234
a₂₆ = 248
Therefore, the value of a₂₆ is 248.
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