The value of a that makes u parallel to v is a = 4 and the value of a that makes u perpendicular to v is -7.
How to solve vectors using dot and scalar product?a. For two vectors to be parallel, their direction ratios must be equal. In other words, their components must have the same ratio. For vectors u = (a, 7) and v = (4, 4), we have:
u1/u2 = a/7 = v1/v2 = 4/4
So the value of a that makes u parallel to v is a = 4.
b. For two nonzero vectors to be perpendicular, their dot product must be equal to zero. The dot product of u = (a, 7) and v = (4, 4) is given by:
u · v = a * 4 + 7 * 4 = 4a + 28
To make this equal to zero, we can solve for a:
4a + 28 = 0
4a = -28
a = -7
So the value of a that makes u perpendicular to v is -7.
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How can I find the perimeter of rectangle
Let's consider what the question asks for:
--> perimeter of a rectangle
To find the perimeter of the rectangle:
--> we need to know the side length
Let's consider how to find our side length:
--> in a rectangle
--> opposite parallel sides are equal in length
--> in a mathematical equation, we get:
[tex]4x-4y=3x+5y\\2-2y=x-3y[/tex]
Now we notice,
--> in the second equation, there is only one 'x'
--> therefore if we find a y-value to substitute into the 'x'
--> we can solve
Let's use the first equation to see how 'x' and 'y' relate to each other:
[tex]4x-4y=3x+5y\\4x-3x=4y+5y\\x=9y[/tex]
Let's use that x-value and plug it the second equation:
[tex]2-2y=x-3y\\2-2y=(9y)-3y\\2-2y=6y\\2=8y\\\\y=\dfrac{1}{4}[/tex]
Since x = 9y:
[tex]x=9y=9*\dfrac{1}{4} =\dfrac{9}{4}[/tex]
Let's find each side length:
[tex]4x-4y=4(\dfrac{9}{4}) -4(\dfrac{1}{4} )=9-1=8\\\\2-2y=2-2(\dfrac{1}{4} )=2-\dfrac{1}{2} =\dfrac{3}{2} =1.5\\\\3x+5y=3(\dfrac{9}{4})+5(\dfrac{1}{4} )=\dfrac{27}{4} +\dfrac{5}{4} =\dfrac{32}{4}=8\\ \\x-3y=\dfrac{9}{4} -3(\dfrac{1}{4})=\dfrac{9}{4}-\dfrac{3}{4} =\dfrac{6}{4} =1.5[/tex]
Let's add up the side length to find the perimeter:
[tex]\text{Perimeter}=8+1.5+8+1.5=9.5+9.5=19[/tex]
Answer: 19
Find the equation of the line intersecting the graph of [tex]y=x^{3} - x+4[/tex] at x=-2 and x = 2
The equation of the line secant to the cubic equation y = x³ - x + 4 is equal to y = 3 · x + 4.
How to derive the equation of a line secant to a curve
In this problem we find the case of a cubic equation that is intersected twice by a line, that is, a secant line. According to analytical geometry, lines are described by equations of the form:
y = m · x + b
Where:
m - Slopeb - Interceptx - Independent variable.y - Dependent variable.Where the slope of the line is determined by secant line formula:
m = Δy / Δx
First, determine the slope of the secant line:
x = - 2
y = (- 2)³ - (- 2) + 4
y = - 2
x = 2
y = 2³ - 2 + 4
y = 10
m = [10 - (- 2)] / [2 - (- 2)]
m = 3
Second, calculate the intercept of the linear function:
b = y - m · x
b = 10 - 3 · 2
b = 10 - 6
b = 4
Third, write the equation of the secant line:
y = 3 · x + 4
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A teacher randomly chooses a two-person leadership team from a group of four qualified students. Three of the students, Sandra, Marta, and Jane, are girls. The fourth student, Franklin, is a boy.
Using the sample space of possible outcomes listed below, where each student is represented by the first letter of his or her name, answer each of the following questions.
What is
�
(
�
)
P(A)P, left parenthesis, A, right parenthesis, the probability that the first student is a boy?
What is
�
(
�
)
P(B)P, left parenthesis, B, right parenthesis, the probability that the second student is a girl?
What is
�
(
�
and
�
)
P(A and B)P, left parenthesis, A, start text, space, a, n, d, space, end text, B, right parenthesis, the probability that the first student is a boy and the second student is a girl?
P(A) = 1/2, P(B) = 3/4, and P(A and B) = 3/8, where A is the event that the first student is a boy and B is the event that the second student is a girl.
The total number of possible outcomes in this scenario is 4C2, which is equal to 6. These outcomes are AB, AC, AD, BC, BD, and CD, where A represents Franklin and B, C, and D represent Sandra, Marta, and Jane, respectively.
The probability that the first student is a boy is P(A) = 1/2, since there are two boys and four students total.
The probability that the second student is a girl is P(B) = 3/4, since there are three girls and four students total.
The probability that the first student is a boy and the second student is a girl is P(A and B) = 1/2 x 3/3 = 3/8, since the probability of the first student being a boy is 1/2 and the probability of the second student being a girl is 3/4 (after one girl has already been chosen as the first student).
Therefore, the probability that the first student is a boy is 1/2, the probability that the second student is a girl is 3/4, and the probability that the first student is a boy and the second student is a girl is 3/8.
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Which of the following is the square of a binomial
The following data gives real GDP (Y), total factor productivity (A), and Capital (K), for two years.
1980 Y=400 A=1 K=25
1990 Y=990 A=1.1 K=81
Assume the production function is Y=A K0.5L0.5. By what percentage did labor grow between 1980 and 1990?
A) 40.75% B) 20.15% C)96% D) 56.25%
The labor grew by 25% between 1980 and 1990. The answer is D) 56.25%
How to calculate the percentages?
Percentage is a way of representing a fraction of 100. To calculate the percentage of a certain quantity, you can use the following formula:
Percentage = (part / whole) x 100
Given the production function Y = A * K⁰ॱ⁵ * L⁰ॱ⁵, we can calculate the growth of labor (L) between 1980 and 1990 by using the information on real GDP (Y), total factor productivity (A), and capital (K).
We can rearrange the production function to find L:
L = (Y / A) / K⁰ॱ⁵
For 1980:
L = (400 / 1) / 25⁰ॱ⁵ = 2
For 1990:
L = (990 / 1.1) / 81⁰ॱ⁵ = 2.5⁰ॱ⁵
Now, we can find the percentage growth of labor between 1980 and 1990 by using the formula:
Percentage growth = (L1990 - L1980) / L1980 * 100
Percentage growth = (2.5 - 2) / 2 * 100 = 25%
So, the labor grew by 25% between 1980 and 1990. The answer is D) 56.25%
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Which expression is equivalent to (1 + cos(x))2Tangent (StartFraction x Over 2 EndFraction) )?
The expression that is equivalent to (1 + cos(x))2Tangent (StartFraction x Over 2 EndFraction) ) is option D. (1 + cos(x))(sin (x))
How are the expressions equivalent?The expression (1 + cos(x))2Tangent (StartFraction x Over 2 EndFraction) is equivalent to (1 + cos(x))(sin (x)) because of the double angle identity for tangent.
The double angle identity states that tangent of 2 times an angle is equal to 2 times the tangent of that angle divided by 1 minus the square of the tangent of that angle. In other words,
tan(2θ) = 2tan(θ)/(1 - tan2(θ))
In this expression, we have tangent of x/2, so substituting θ = x/2 gives us:
tan(x) = 2tan(x/2)/(1 - tan2(x/2))
Since cos(x) = 1 - 2sin2(x/2), we can simplify the expression to:
(1 + cos(x))2tan(x/2) = (1 + 1 - 2sin2(x/2))2tan(x/2) = (2 - 2sin2(x/2))(2sin(x/2)/(1 - sin2(x/2)))
Expanding the product of the two factors gives us the final result:
(1 + cos(x))2tan(x/2) = (2 - 2sin2(x/2))(sin(x)) = (1 + cos(x))(sin(x))
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equation 3x+y=0. Plot the theme points that are in the solution wet, drow a line
through the three points and then answer the questions below
(1,7) (-8,4) (-2,6) (-5,5) (2,-6) (3,-9)
Step-by-step explanation:
The equation 3x + y = 0 can be rearranged to y = -3x. This means that the graph of the equation is a straight line with slope -3 and y-intercept of 0.
The three points that are in the solution set are (-2,6), (-5,5), and (3,-9). Plotting these points on a coordinate plane and connecting them with a straight line, we can see that they all lie on the same line with slope -3:
[Graph not shown here, but a straight line with three points (-2, 6), (-5, 5), (3, -9) can be plotted with a slope of -3 and y-intercept of 0.]
From the graph, we can see that the line passes through the origin (0,0), which means that the y-intercept is 0. The slope of the line is -3, which means that the line has a negative slope and is downward sloping. The line separates the plane into two half-planes, one above the line and one below the line. The points above the line have y-coordinates greater than 0 and the points below the line have y-coordinates less than 0.
Suppose Sine (x) = negative three-fifths and cos(x) < 0. What the value of cos(2x)?
The value of cos (2x) is 7/25.
What are Trigonometric Functions?Trigonometric functions are defined as the real functions which are simply the functions of an angle of a triangle. They are basically the periodic functions which relate an angle in a right angled triangle to the ratios of the length of two sides.
Given that,
sin x = -3/5 and cos (x) < 0
We have a trigonometric formula,
cos (2x) = 1 - 2 sin²(x)
Substituting the values given,
cos (2x) = 1 - 2 × (-3/5)²
= 1 - (2 × 9/25)
= 1 - 18/25
= 7/25
Hence the value of cos (2x) is 7/25.
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Consider Functions: Consider a square with side of length s, diagonal of length d, perimeter P, and area A.a) Write A as a function of s.b) Write s as a function of A.c) Write s as a function of d.d) Write d as a function of s.e) Write P as a function of s.f) Write s as a function of P.g) Write A as a function of P.h) Write d as a function of A.
For square, Functions will be A=s², s=√A, s=√(d²/2), d=√2s², p=4s, s=p/4, A=(p/4)², d=√2A.
What exactly is a function?
A function is defined as a relationship between a group of inputs that each have one output. A function is a connection between inputs in which each input is associated to exactly one output. Every function has a domain and a co-domain, as well as a range. In general, a function is denoted as f(x), where x represents the input. A function's generic representation is y = f. (x).
In mathematics, there are several types of functions. Some examples include:
When there is a mapping for a range for each domain between two sets, this is referred to be an injective function or a one to one function.
Surjective functions, also known as Onto functions, are used when more than one element is transferred from domain to range.
Polynomial function: A function made up of polynomials.
Inverse Functions: A function that may be used to inverse another function.
Now,
As given square with side of length s, diagonal of length d, perimeter P, and area A.
and Area=side²
Perimeter=4*side
diameter²=side²+side²
then A=s² and s=√A, s=√(d²/2) and d=√2s², p=4s and s=p/4, A=(p/4)², d=√2A.
Hence,
For square, Functions will be A=s², s=√A, s=√(d²/2), d=√2s², p=4s, s=p/4, A=(p/4)², d=√2A.
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Verify that the indicated function y p(x) is an explicit solution of the given first-order differential equation.
(y-x)y' =y-x+ 2; y=x+2√x+3
When y = x + 2√x +3,
y'= -x+2
Thus, in terms of x,
(y - x)y' =
y-x+2=
Since the left and right hand sides of the differential equation are equal when x + 2√x + 3 is substituted for y, y = x + 2√x + 3 is a solution.
Proceed as in Example 6, by considering p simply as a function and give its domain. (Enter your answer using interval notation.)
Then by considering p as a solution of the differential equation, give at least one interval I of definition.
O(-6, -3)
O(-3,00)
(-∞, -3)
x.
(-6, 3)
O[-3, 3]
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[tex]y=\frac{2x\pm\sqrt{5x^2-(4x^2-16x+4\bar{c})} }{2(1)}[/tex]Solve the given DE, [tex](y-x)\frac{dy}{dx} =y-x+2[/tex].
Rewriting,
=> [tex](y-x)\frac{dy}{dx} =y-x+2[/tex]
=> [tex](y-x)dy =(y-x+2)dx[/tex]
=> [tex]-(y-x+2)dx+(y-x)dy =0[/tex]
=> [tex](-y+x-2)dx+(y-x)dy =0[/tex]
Check to see if this is an exact DE by taking the partial derivative of M with respect to y and N with respect to x.
[tex]M=(-y+x-2)dx[/tex]
=> [tex]M_{y} =-1[/tex]
[tex]N=(y-x)dy[/tex]
=> [tex]N_{x}=-1[/tex]
[tex]M_{y} =N_{x}[/tex], so this is an exact DE. Now integrate M with respect to x and N with respect to y.
[tex]\int\ ({-y+x-2)} \, dx[/tex]
=>[tex]-xy+\frac{x^2}{2}-2x[/tex]
[tex]\int\ ({y-x)} \, dy[/tex]
=> [tex]=\frac{y^2}{2} -xy[/tex]
So we can say the solution to the given DE is, [tex]\frac{x^2}{2}+\frac{y^2}{2}-xy-2x=c[/tex].
Rosa is using a recipe that serves six and uses one and three quarters cups of pasta. Choose the amount of pasta she will use if she wants to make eight servings. (2 points)
two and one third cups
three and three quarters cups
four and one quarter cups
two and one quarter cups
Answer: To calculate the amount of pasta needed for eight servings, we need to multiply the original amount of pasta in the recipe by 8/6.
So, 1.75 cups * 8/6 = 2.3 cups of pasta.
Therefore, Rosa will use 2.3 cups or four and one quarter cups of pasta if she wants to make eight servings.
Step-by-step explanation:
HELP ASAP, TELL ME THE CODE FOR THIS STUFF (for example, AGHI also it has to be in all caps.
Volume of Cylinders, Cones, and Spheres
1
2
The diameter of a sphere is 4 centimeters. Which represents the volume of the sphere?
○ 3² cm³
π
O 87 cm³
○ 64 cm³
O 167 cm³
Answer:[tex]\frac{32\pi }{3} cm^{3}[/tex]
Step-by-step explanation:
Since the diameter is 4 cm, we know the radius is 2 cm since diameter = 2 x radius. The formula for a sphere's volume is [tex]\frac{4\pi }{3}[/tex]×[tex]r^{3}[/tex], so by plugging in 2 we get [tex]\frac{4\pi }{3}[/tex]×[tex]2^{3}[/tex] = [tex]\frac{4\pi }{3}[/tex] x 8 = [tex]\frac{32}{3}[/tex] [tex]\pi[/tex] [tex]cm^{3}[/tex]
when the stretched string of the apparatus represented below is made to vibrate, point p does not move. point p is most probably at the location of
A node is a point of no displacement in a standing wave. Therefore, if point P does not move, it is most likely located at a node. the points along the wave that experience maximum displacement are called antinodes.
A node is a point of no displacement in a standing wave, meaning that if the stretched string of the apparatus represented is made to vibrate, point P will not move. Point P is most likely located at a node as it experiences no displacement. A standing wave is created when two waves combine and the resulting wave is stationary. The points along the wave that experience no displacement are called nodes, and the points along the wave that experience maximum displacement are called antinodes. Nodes can be found at points that are integral multiples of half the wavelength of the wave. Therefore, it can be concluded that point P is at a node since it does not move when the string is made to vibrate.
The complete question is :
When the stretched string of the apparatus represented below is made to vibrate, point p does not move. Point p is most probably at the location of _____.
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10. Five men took 15 days to dig a hole. How long would 25 men take to dig the same hole working at the same pace?
Answer: Three days
5 men divided by the 15 days it took to dig the hole, equals 3. Which means it took three times as much time as the amount of workers who dug the hole.
This means it will take three days to dig the whole with 25 workers.
I hope this helped. Good Luck <3.
Your goal is to have $17500 in your bank account by the end of 9 years. If the interest rate remains constant at 4% and you want to make annual identical deposits, how much will you need to deposit in your account at the end of each year to reach your goal? a. 1984.36
b. 1653.63
c. 1157.54
d. 1488.27
The annual deposits that needs to be made is for amount $2352.94.
What is annuity?
Annuity refers to an equal series of future cash flows which are received or paid periodically. Future value of annuity is the value of the annuity at the end of the series whereas present value is the value of the annuity at the beginning of the series.
The annuity value can be calculated as -
Future value of annuity = Annuity x (1 - (1 + Rate)^-Number of years) / Rate
The values are given as -
Future value of annuity = $17500.00
Annuity = Identical annual deposits
Rate = 4% = 0.04
Number of years = 9
Substitute the values into the equation -
17500 = Annuity x (1 - (1 + 0.04)^-9 ) / 0.04
Annuity identical deposits = 17500 x 0.04 / ((1.04)^-9)
Annuity identical deposits = 700 / (1 - 0.7025)
Annuity identical deposits = 700 / 0.2975
Annuity identical deposits = 2352.94
Therefore, the value is obtained as $2352.94.
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Construct a know-show table for each of the following statements and then write a formal proof for one of the statements.
(a) If m is an odd integer, then m + 1 is an even integer.
(b) If x is an even integer and y is an odd integer, then x + y is an odd integer
(c) If m is an even integer, then 3m^2 + 2m + 3 is an odd integer.
Step-by-step explanation:
(a)
If m is an odd integer, then m + 1 is an even integer.
m (odd integer) m + 1 (even integer)
1 2
3 4
5 6
... ...
Proof:
Suppose m is an odd integer. We can write m as 2n + 1 for some integer n. Then,
m + 1 = (2n + 1) + 1 = 2n + 2
Since 2n + 2 is clearly an even integer, it follows that m + 1 is an even integer if m is an odd integer.
(b)
If x is an even integer and y is an odd integer, then x + y is an odd integer
x (even integer) y (odd integer) x + y (odd integer)
0 1 1
2 3 5
4 5 9
... ... ...
Proof:
Suppose x is an even integer and y is an odd integer. We can write x as 2n and y as 2m + 1 for some integers n and m. Then,
x + y = 2n + (2m + 1) = 2(n + m) + 1
Since n + m is clearly an integer, it follows that x + y is an odd integer if x is an even integer and y is an odd integer.
(c)
If m is an even integer, then 3m^2 + 2m + 3 is an odd integer.
m (even integer) 3m^2 + 2m + 3 (odd integer)
0 3
2 27
4 99
... ...
Proof:
Suppose m is an even integer. We can write m as 2n for some integer n. Then,
3m^2 + 2m + 3 = 3(2n)^2 + 2(2n) + 3 = 12n^2 + 4n + 3
Since 12n^2 + 4n + 3 is clearly an odd integer, it follows that 3m^2 + 2m + 3 is an odd integer if m is an even integer.
Please helpppppppp meeeeee asappppppppp pleaseeeeeeee !!???
To find a common denominator for 7/8 and 13/16, we can find the least common multiple (LCM) of 8 and 16, which is 16.
LCM's meaning ?
lowest common factor
Describe LCM. Least Common Multiple is a mathematical term. The smallest number that is a multiple of both of two numbers is called the least common multiple.
7/8 can be written as 7 * (2/2) / 8 = 7 * 2 / 16 = 7/16
13/16 is already in the form of a fraction with a denominator of 16, so no further modification is needed.
So, 7/8 can be written as 7/16 and 13/16 can be written as 13/16.
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evaluate the integral below by interpreting it in terms of areas in the figure. the areas of the labeled regions are
The integral evaluates to 11, which is the sum of the areas of the three regions (R1 + R2 + R3 = 4 + 5 + 6 = 11).
R1 = 4, R2 = 5, R3 = 6
The integral evaluates to 11, which is the sum of the areas of the three regions (R1 + R2 + R3 = 4 + 5 + 6 = 11).
The integral is given by:
∫ (R1 + R2 + R3) dA
where R1, R2, and R3 are the areas of the labeled regions in the figure.
By interpreting the integral in terms of areas, we can calculate the value of the integral. The integral evaluates to 11, which is the sum of the areas of the three regions (R1 + R2 + R3 = 4 + 5 + 6 = 11).
The complete question is :
Evaluate the integral below by interpreting it in terms of areas in the figure. The areas of the labeled regions are A = 3, B = 4, C = 5, and D = 6.
∫DBCA x dA
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A company believes it has a 40% chance of being successful on bidding a contract that yields a net profit of $30,000. Assume it costs $5,000 in consultant fees to prepare the bid. What is the expected gain or loss for the company if it decides to bid on the contract?
Answer:
Joe mum so gaeee
Step-by-step explanation:
your answer is 1000
Write an equation for the quadratic graphed below
x-intercepts: (-2,0) and (1,0). y-intercept: (0,-2)
A quadratic function with x-intercepts as (-2,0) and (1,0) and y-intercept as (0,-2) is y = x² + x - 2.
What is a quadratic function?
A polynomial function with one or more variables, where the largest exponent of the variable is two, is referred to as a quadratic function. It is also known as the polynomial of degree 2 since the greatest degree term in a quadratic function is of second degree.
It is given that the x intercepts of the quadratic function are (-2,0) and (1,0).
The y intercept of the quadratic function is (0,-2).
Let the equation of the given quadratic be y = ax² + bx + c.
As the given quadratic has x intercepts as (-2,0) and (1,0) and y intercept as (0,-2).
This implies that the quadratic function is passing through the points (-2,0), (1,0) and (0,-2).
So, the points (-2,0), (1,0) and (0,-2) must satisfy the equation of the quadratic function.
As the point (-2,0) satisfy the equation of the quadratic function -
y = ax² + bx + c
0 = a(-2)² + b(-2) + c
0 = 4a - 2b + c ..... (1)
As the point (1,0) satisfy the equation of the quadratic function -
y = ax² + bx + c
0 = a(1)² + b(1) + c
0 = a + b + c ..... (2)
As the point (0,-2) satisfy the equation of the quadratic function -
y = ax² + bx + c
-2 = a(0)² + b(0) + c
-2 = c ..... (3)
Substitute the value of c in equation (1) -
0 = 4a - 2b - 2
2 = 4a - 2b ...... (4)
Substitute the value of c in equation (2) -
0 = a + b - 2
2 = a + b ...... (5)
Multiply equation (5) by 2 -
4 = 2a + 2b ...... (6)
Add equation (4) and (6) -
2 + 4 = 4a - 2b + 2a + 2b
6 = 6a
a = 1
Substitute the value of a in equation (5) -
2 = 1 + b
b = 2 - 1
b = 1
The values are a = 1, b = 1 and c = -2.
Now substitute the value of a, b and c in the quadratic function.
y = ax² + bx + c
y = (1)x² + (1)x + (-2)
y = x² + x - 2
Therefore, the quadratic function is y = x² + x - 2.
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Given the preimage ABC, describe a sequence of transformations that produces the
image A'B'C'
Answer:
At point B rotate ABC clockwise 90 degrees. Translate the figure two units to the left. At point B dilate the figure with a scale factor of 2.
Step-by-step explanation:
A random sample of 223 students were asked if they owned a pet or not. The following contingency table gives the two-way classification of their responses.
The probabilities of the random sample of 223 students are solved
P ( male ) = 0.475
P ( female ) = 0.525
P ( male | pet ) = 0.538
P ( female | no pet ) = 0.547
What is Probability?The probability that an event will occur is measured by the ratio of favorable examples to the total number of situations possible
Probability = number of desirable outcomes / total number of possible outcomes
The value of probability lies between 0 and 1
Given data ,
Let the total number of students be = 223 students
Let the total number of male students be = 49 + 57 = 106 students
Let the total number of female students be = 64 + 53 = 117 students
Now , the equation will be
Let the number of male students who own a pet = 57 students
Let the number of male students who does not own a pet = 49 students
And ,
Let the number of female students who own a pet = 53 students
Let the number of female students who does not own a pet = 64 students
The probability of choosing a male student P ( male ) = number of male students / total number of students
The probability of choosing a male student P ( male ) = 106 / 223
The probability of choosing a male student P ( male ) = 0.475
And ,
The probability of choosing a female student P ( female ) = number of male students / total number of students
The probability of choosing a female student P ( female ) = 117 / 223
The probability of choosing a female student P ( female ) = 0.525
And ,
Probability of choosing a male student who owns a pet P ( male | pet ) = number of male students who own a pet / number of male students
P ( male | pet ) = 57 / 106
P ( male | pet ) = 0.538
The probability of choosing a female student who does not own a pet is P ( female | no pet ) = number of female students who does not own a pet / number of female students
P ( female | no pet ) = 64 / 117
P ( female | no pet ) = 0.547
Hence , the probabilities are solved
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CAN SOMEONE HELP WITH THIS?✨
4966.5
Step-by-step explanation:
Starting at finding out how the population will increase in 3 years we take 3 and divide it by 4. This produces an increase of 75% every 3 years. If we multiply 2838 by 75% we get 2128.5. If we add it back to 2838, we get 4966.5
A spherical boulder is 24 feet in diameter and weighs almost 6 tons find the volume
The volume of the spherical boulder is 7234.56 cubic feet.
What is the diameter?
A line connecting the center and the circumference at its opposite ends is called the diameter. Its length is double that of the circle's radius.
The formula for the volume of a sphere is [tex]V=\frac{4}{3}\pi r^{3}[/tex].
Given the diameter of the sphere is 24 feet.
therefore radius is equal to 12 feet.
The volume of the sphere is equal to
[tex]V=\frac{4}{3}\pi (12)^{3} \\V=\frac{4}{3} *3.14*1728\\V=7234.56[/tex]
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If the lengths of two adjacent sides of a parallelogram area a and b, and if the acute angle formed by these two sides is theta, show that the product of the lengths of the two diagonals is given by the expression (a^2 + b^2)^2 - 4a^2b^2cos^2theta
√(a² + b²)² - 4a²b²cos²θ is the product of the lengths of the two diagonals is given by the expression.
What is a mathematical expression?
A mathematical expression is a phrase that includes at least two numbers or variables, at least one arithmetic operation, and the expression itself. This mathematical operation may be addition, subtraction, multiplication, or division.
An expression's structure is as follows: Number/variable, Math Operator, Number/Variable is an expression.
we have AB as a, AD as b and the angle between them is theta.
So using the cosine rule, we have
BD = √a² + b² - 2abcosθ
So now consider the triangle ABC
Here AB is a, BC is b and the angle is 180-theta
So using cosine rule, we get AC as
AC = √a² + b² - 2abcosθ( 180 - θ )
AC = √a² + b² - 2ab(-cosθ )
AC = √a² + b² - 2abcosθ
Now we have the two diagonals AC and BD. So multiplying, we get
AC × BD = √a² + b² + 2abcosθ × √a² + b² - 2abcosθ
Simplifying, we get
AC × BD = √(a² + b² + 2abcosθ) × (√a² + b² - 2abcosθ)
AC × BD = √(a² + b²)² - (2abcosθ)²
AC × BD = √(a² + b²)² - 4a²b²cos²θ
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I’m not sure how to answer these questions can someone help pls
Answer:
Greatest common factor for 14/16 is 2
Greatest common factor for 3/12 is 3
Greatest common factor for 16/28 is 4
So to simplify 14/16 it is 7/8
To simplify 3/12 it is 1/4
To simplify 16/28 it is 4/7
Step-by-step explanation:
Given that 6^(y+3)=2(x+9), find the ratio of x to y
The ratio of x to y is given by 3 : 1
What is Proportion?The proportion formula is used to depict if two ratios or fractions are equal. The proportion formula can be given as a: b::c : d = a/b = c/d where a and d are the extreme terms and b and c are the mean terms.
The proportional equation is given as y ∝ x
And , y = kx where k is the proportionality constant
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Given data ,
Let the proportion be represented as A
Now , the value of A is
6 ( y + 3 ) = 2 ( x + 9 )
On simplifying , we get
6y + 18 = 2x + 18
Subtracting 18 on both sides , we get
6y = 2x
Divide by 2 on both sides , we get
x = 3y
Divide by 3 on both sides , we get
x/y = 3/1
Therefore , the proportion is x : y : : 3 : 1
Hence , the ratio is 3 : 1
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PLEASE HELP ME!
Anna is considering writing and publishing her own book She estimates her revenue equation as R = 6.56x and her cost equation as C = 10.063 + 1.09x where x is the number of books she sells. Find the minimum number of books she must sell to make a profit
Anna must sell atleast ? books to make a profit.
Anna must sell at least approximately 1.845 books to make a profit.
What is the linear equation?
A linear equation is an algebraic equation of the form y=mx+b. where m is the slope and b is the y-intercept.
We can find the minimum number of books Anna must sell to make a profit by setting the revenue equal to the cost and solving for x.
That is, we want to find the value of x where R = C:
6.56x = 10.063 + 1.09x
5.47x = 10.063
x = 10.063 / 5.47
x = approximately 1.845 books
Hence, Anna must sell at least approximately 1.845 books to make a profit.
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Write the decimal number in words 17.43