Answer: [6, 25]
Explanation:
If Gerald takes a 700-mile round trip, then he is credited with 1400 miles flown (700 miles each way). To attain Gold status, Gerald needs to fly between 8400 and 35000 miles per year. Let's call the number of round trips Gerald needs to take "x". Then, we can set up an inequality to represent the number of miles he needs to fly:
8400 ≤ 1400x ≤ 35000
Dividing all three parts of the inequality by 1400, we get:
6 ≤ x ≤ 25
Therefore, Gerald needs to visit his parents at least 6 times a year, but no more than 25 times a year to attain Gold status. In interval notation, we can express this as:
[6, 25]
The triangles below are similar. Find the length of x.
Answer:
D
Step-by-step explanation:
If the triangles are similar, their side ratios are equal
[tex] \frac{54}{12} = \frac{x}{15} [/tex]
Cross-multiply to find x:
[tex]x = \frac{54 \times 15}{12} = 67.5[/tex]
Write a function based on the given parent function and transformations in the given order.
Parent function:
() Shift units to the left.
() Stretch horizontally by the factor of .
() Reflect across the -axis.
The function based on the given parent function and transformations in the given order would be:
f(x)= -1/2√(x+2)
What is a function?Functions are often represented as equations, graphs, or tables, and they are used to model a wide range of real-world phenomena. For example, a function might describe the relationship between the time of day and the temperature outside, or the amount of money earned based on the number of hours worked.
Let's break down how each transformation affects the parent function:
Shifting units to the left: This transformation involves replacing the variable "x" with "x+2" in the parent function. This has the effect of shifting the graph 2 units to the left. The function becomes:
f(x)= √(x+2)
Stretching horizontally: This transformation involves replacing "x" with "2x" in the function obtained after the first transformation. This has the effect of stretching the graph horizontally by a factor of 1/2. The function becomes:
f(x)= √(2x+4)
Reflecting across the x-axis: This transformation involves multiplying the entire function by -1. This has the effect of reflecting the graph across the x-axis. The function becomes:
f(x)= -√(2x+4)
So the final function based on the given parent function and transformations in the given order is:
f(x)= -1/2√(x+2)
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20% OF a IS 15. WHAT IS a
Answer:
The answer is 3
Step-by-step explanation:
20% of 15 can be written as 20% × 15
= 20/100 × 15
= 3
Thus, 20% of 15 is 3.
hope this helps
Does the function model exponential growth or decay? g ( t ) = 1.7 ⋅ 0. 8 t
The answer of the given question based on function is g(t) = 1.7 * 0.8^t models exponential decay.
What is Exponential decay?Exponential decay is a type of mathematical function that represents a process in which the value of a quantity decreases over time or through a series of events at a constant proportional rate. In exponential decay, the rate of decrease is proportional to the current value of the quantity.
Exponential decay is often modeled by the following equation:
y = a * e^(-kt)
The function g(t) = 1.7 * 0.8^t models exponential decay.
In this function, the base of the exponent is 0.8, which is a number between 0 and 1. When a base of an exponential function is between 0 and 1, it represents exponential decay. This means that as the value of t increases, the value of g(t) decreases at an increasingly rapid rate.
Additionally, the coefficient of the exponent (1.7 in this case) represents the initial value of the function. Since the coefficient is positive, the initial value is also positive. Therefore, the function represents exponential decay from a positive initial value.
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A company's daily profit from the production and sale of electrical components can be described by the equation P(x)=6.30x-1,777 dollars, where x is the number of units produced and sold. What level of production and sales will give a daily profit of more than $8,9337
Answer:1,777...)-(1,555)= B)1,333...+4,666
Step-by-step explanation:1,777...)-(1,555)= B)1,333...+4,666
Solve for d in the equation below. Isolate [tex]d[/tex] on the left hand side. Show all work.
[tex]A=B\left(1+\dfrac{c}{d} \right)^{dk}[/tex]
The solution for ‘d’ in the given equation is d = (ln(A) - ln(B))/ln(1+c/d).
What is natural log?Natural log is a logarithmic function commonly used in mathematics and in calculators to simplify equations. It is the inverse of the exponential function and is written as ln(x).
In order to solve for ‘d’ in the given equation, we need to isolate it on the left side of the equation.
To do this, we can begin by taking the natural log of both sides of the equation, as shown below.
ln(A) = ln(B) + d*ln(1+c/d)
We can now rearrange the terms in this equation such that ‘d’ is isolated on the left side of the equation.
d*ln(1+c/d) = ln(A)-ln(B)
We can now divide both sides of the equation by ln(1+c/d) in order to solve for ‘d’, as shown below.
d = (ln(A) - ln(B))/ln(1+c/d)
Therefore, the solution for ‘d’ in the given equation is
d = (ln(A) - ln(B))/ln(1+c/d).
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Will be brainliest
. If P = 2x² + 3xy-5y², Q=-5x² +2xy + 3y² and R=-3x² + 5xy - 2y^2, show that P+Q-R=0.
Answer:
Step-by-step explanation:
[tex]P+Q-R=(2x^2+3xy-5y^2)+(-5x^2+2xy+3y^2)-(-3x^2+5xy-2y^2)[/tex]
[tex]=2x^2+3xy-5y^2-5x^2+2xy+3y^2+3x^2-5xy+2y^2[/tex]
[tex]=2x^2-5x^2+3x^2-5y^2+3y^2+2y^2+3xy+2xy-5xy[/tex]
[tex]=0[/tex]
Answer:
To show that P + Q - R = 0, we need to substitute the expressions for P, Q, and R into the equation and simplify.
P + Q - R = (2x² + 3xy - 5y²) + (-5x² + 2xy + 3y²) - (-3x² + 5xy - 2y²)
Simplifying the brackets on the right-hand side:
P + Q - R = 2x² + 3xy - 5y² - 5x² + 2xy + 3y² + 3x² - 5xy + 2y²
Grouping like terms together:
P + Q - R = (2x² - 5x² + 3x²) + (3xy + 2xy - 5xy) + (-5y² + 3y² + 2y²)
Simplifying the expressions in the brackets:
P + Q - R = 0x² + 0xy + 0y²
Therefore, P + Q - R = 0.
Question 10
With the points A(6, 2) B(-4,-4)
C(-8, 12) D(2, -8).
What are the new points if the scale factor of dilation is 1/2?
Answer:
A'(3, 1); B'(-2, 2); C'(-4, 6); D'(1, -4)-----------------------------------
The scale factor k affects the points with the rule:
(x, y) → (kx, ky)Apply the rule to the given points, given k = 1/2:
A(6, 2) → A'(3, 1);B(-4, 4) → B'(-2, 2);C(-8, 12) → C'(-4, 6);D(2, -8) → D'(1, -4)Omaya picked as amount of apples, took a break, and then picked v more. Write the expression that models the total number of apples Omaya picked
The formula that describes how many apples Omaya picked in total is Y=x + v.
What does a linear equation mean in mathematics?
A linear equation is one that only has a constant and a first-order (linear) component, like y=mx+b, where m is the slope and b is the y-intercept.
Occasionally, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.
The term "linear equation" refers to equations with power 1 variables. Where ax+b = 0 is one example with only one variable, where a and b are real numbers and x is the variable.
Let Total number of apples= Y
so,
Y=x + v
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calculate the following expression and give answer in the standard form z=a+bi
use the De Moivre's theorems to calculate
[tex] \frac{(cis22.5)(5cis67.5)}{3 {}^{2} ( \cos(12) + i \sin(12)) } [/tex]
As a result, the expression can be expressed in the following standard form: -32 sin(102) + 32i cos = 32i (cos(102) + I sin(102)) (102).
what is expression ?In mathematics, programming, or other subjects, an expression is typically a grouping of numbers, symbols, and operators that denotes a value or computation. When used to indicate a mathematical relationship or formula, an expression in mathematics can be made up of a number of different variables, operators, and digits. For instance, the statement 3x + 2y denotes a linear equation. A combination of variables, constants, operators, and functions can make up an expression in programming, which can then be evaluated to create a value. For instance, the expression 5 + 3 evaluates to 8 as an expression.
given
First, we can use De Moivre's theorem to simplify the expression:
(cis(22.5+67.5)) = (cis(22.5+67.5)) = cis(22.5+90) = I
We then have:
Cos(12) + I sin(12) = 32cis (12)
When we multiply these two outcomes, we obtain:
32i cis(90+12) = 32i cis(i * 32cis(12)) = 32i cis102
Thus, the phrase can be worded as follows:
(cis22.5) (5cis67.5) (5cis67.5) 32(cos (12) + I sin (12)) = 32i cis (102).
Using Euler's formula, cis102 can be written as follows:
102 + I sin + cos(102) = cis102 (102)
As a result, the expression can be expressed in the following standard form: -32 sin(102) + 32i cos = 32i (cos(102) + I sin(102)) (102)
The final response is thus: Z = 32i cos + 32sin(102) (102)
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Which of the following is the best estimate of the area of the irregular shape?
*The answer is not 19.5*
Option (D) 15.5 units²
Step-by-step explanation:Number of clear squares = 9 = 9 squares
Number of half squares = 7 = 3.5 squares
Number of more than half squares = 1 = 1 squares
Number of less than half squares = 8 = 2 squares
Grand total = 9 + 3.5 + 1 + 2
= 15.5 units²
Please mark my answer as the BRAINLIEST. Plssssssss
Hope my answer helps:)The following graph depicts which inverse trigonometric function
It should be noted that cpnsidering that when x = 1, y = 0, it is found that the inverse trigonometric function depicted in the graph is given by: A. y = Arccos x
What is an inverse trigonometric function?It has the format y = f(x), with f being an inverse trigonometric function, and it basically is what angle y has a trigonometric function value of x.
In this graph, we have that when x = 1, y = 0, and considering that cos(0) = 1, the inverse trigonometric function depicted in the graph is given by y = Arccos x
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what is 0.357 divided by 6.7
Answer:
0.05328358208955223880597014925373
Step-by-step explanation:
Show that f(x) can be written as Px2+Qx+R+Vx+3+Wx-5and find the values of P, Q, R, Vand W.
f(x) = (3/2)*x² + (13/2)*x - 5/2 - 6/(x + 3) + 15/(x - 5)
and the values of P = 3/2, Q = 13/2, R = -5/2, V = -6, W = 15
What is the fraction?
A fraction is a number that represents a part of a whole or a ratio between two quantities. It is written with a numerator (the top number) and a denominator (the bottom number) separated by a horizontal line. The numerator represents the number of parts being considered, while the denominator represents the total number of parts that make up a whole unit.
First, let's factor in the denominator of the given function:
x² - 2x - 15 = (x - 5)(x + 3)
Now we can use partial fraction decomposition to express the given function as the sum of simpler fractions:
f(x) = (x⁴ + 2x³ - 29x² - 47x + 77) / (x² - 2x - 15)
= (x⁴ + 2x³ - 29x² - 47x + 77) / ((x - 5)(x + 3))
= (Px² + Qx + R) + V/(x + 3) + W/(x - 5)
where P, Q, and R are constants to be determined, and V and W are constants to be determined as well.
To find the values of P, Q, and R, we can use polynomial long division to divide the numerator by the denominator:
x² + 5x - 2
x² - 2x - 15 | x⁴ + 2x³ - 29x² - 47x + 77
- (x⁴ - 2x³ - 15x²)
----------------------
4x³ - 14x² - 47x
- (4x³ - 8x² - 60x)
------------------
-6x² + 47x
- (-6x² + 12x + 90)
--------------
35x - 90
Therefore,
f(x) = (x⁴ + 2x³ - 29x² - 47x + 77) / (x² - 2x - 15)
= x² + 5x - 2 - (6x² - 35x + 90) / ((x - 5)(x + 3))
= -5/2 + (13/2)x + (3/2)x² + V/(x + 3) + W/(x - 5)
Comparing this to the given form, we can see that:
P = 3/2
Q = 13/2
R = -5/2
V = -6
W = 15
Therefore, we have:
f(x) = (3/2)*x² + (13/2)*x - 5/2 - 6/(x + 3) + 15/(x - 5)
and the values of P = 3/2, Q = 13/2, R = -5/2, V = -6, W = 15
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Ms. grimes walks up to a tank of water that can hold up to 12 gallons. When it is active, a drain empties water from the tank at a constant rate. When jada first sees the tank it contains 8 gallons of water. Four minutes later the tank contains 6 gallons of water. At what rate is the amount of water in the tank changing? Use a signed number, and include the unit of measurement in your answer.
The rate at which the amount of water in the tank is changing is -0.5 gallons/minute. This means that the amount of water in the tank is decreasing at a rate of 0.5 gallons per minute.
Rate of change:
The rate of change is a measure of how a quantity changes with respect to another quantity. It tells us how much a particular quantity is changing for a given change in another quantity.
It is often represented as a ratio of the change in the quantity of interest to the change in the other quantity, usually with units of measurement.
Here we have
Ms. Grimes walks up to a tank of water that can hold up to 12 gallons.
When it is active, a drain empties water from the tank at a constant rate. When Jada first sees the tank it contains 8 gallons of water. Four minutes later the tank contains 6 gallons of water.
We can use the following formula for the rate of change of a quantity:
Rate of change = (change in quantity) / (change in time)
Given that when Jada first saw the tank, 8 gallons. Four minutes later, the amount of water in the tank decreased to 6 gallons.
So the change in the amount of water in the tank is:
change in water = 6 gallons - 8 gallons = -2 gallons
The change in time is:
change in t = 4 minutes - 0 minutes = 4 minutes
So, the rate of change of the amount of water in the tank is:
= (change in W) / (change in t)
= (-2 gallons) / (4 minutes) = -0.5 gallons/minute
Therefore,
The rate at which the amount of water in the tank is changing is -0.5 gallons/minute. This means that the amount of water in the tank is decreasing at a rate of 0.5 gallons per minute.
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Morgan like and newton are 24 miles apart on a map the two cities are 3 inches apart what is the map scale
Answer:
8 mi per in.
Step-by-step explanation:
Divide 24 mi by 3 in. to get base number of 1 in. per x mi
24/3 = 8
8 mi per in.
2. Use the Factor Theorem to determine if (x-4) is a factor of the polynomial function
P(x) = x5 + 4x4 - 17x³ - 60x²
Clearly state that (x-4) is a factor, or that (x-4) is not a factor. Show all work.
Answer:
Yes, (x - 4) is a factor of polynomial function P(x) = x⁵ + 4x⁴ - 17x³ - 60x².
Step-by-step explanation:
The Factor Theorem states that if f(x) is a polynomial, and f(a) = 0, then (x - a) is a factor of f(x).
Therefore, according to the Factor Theorem, if (x - 4) is a factor of P(x) then P(4) = 0.
To determine if (x - 4) is a factor of P(x), substitute x = 4 into the function and solve.
[tex]\begin{aligned}x=4 \implies P(4)&=(4)^5+4(4)^4-17(4)^3-60(4)^2\\&=1024+4(256)-17(64)-60(16)\\&=1024+1024-1088-960\\&=2048-1088-960\\&=960-960\\&=0\end{aligned}[/tex]
As P(4) = 0, this confirms that (x - 4) is a factor of polynomial function P(x) = x⁵ + 4x⁴ - 17x³ - 60x².
[tex]\hrulefill[/tex]
Additional comments
The fully factored form of the given polynomial function is:
[tex]P(x)=x^2\left(x+3\right)\left(x-4\right)\left(x+5\right)[/tex]
If the area of a square equals 121sq inches, what is the perimeter?
Answer: 44
Step-by-step explanation: So, if the area of a square is 121 sq. in, the sides that could be multiplied to get 121 are 11 x 11. Then, you multiply by four to get the perimeter.
13. Find x.
E(7x°
D- (4x+10)°
A-(4x + 5)°
C-(5x - 5)
B-(6x + 10)°
Answer: 14ex
Step-by-step explanation:
How can I solve this?
The polynomial that models the company's profit is 1.0x² + 3.5x, which represents the profit in thousands of dollars.
Define polynomial equationA polynomial equation is an algebraic equation that involves a polynomial, which is an expression consisting of variables and coefficients that are combined using addition, subtraction, multiplication, and exponentiation (raising to a power).
The polynomial that models the company's sales in thousands of dollars is simply the given sales volume polynomial:
Sales = 5.9x² + 8.5x + 4.7 (in thousands of dollars)
To find the polynomial that models the company's profit, we need to subtract the cost polynomial from the sales polynomial:
Profit = Sales - Cost
Profit = (5.9x² + 8.5x + 4.7) - (4.9x²+ 5x + 4.7)
Profit = (5.9 - 4.9)x² + (8.5 - 5)x
Profit = 1.0x²+ 3.5x
So the polynomial that models the company's profit is 1.0x² + 3.5x, which represents the profit in thousands of dollars.
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A bag contains 3 gold marbles, 26 silver marbles, and 28 black marbles. Someone offers to play this game: You randomly select on marble from the bag. If it is gold, you win $3. If it is silver, you win $2. If it is black, you lose $1. What is your expected value if you paid $1 to play this game?
3+26+28 = 57
chance of gold is 3/57
chance of silver = 26/57
chance of black = 28/57
the expected value of playing this game = a weighted average of the winnings (or losses) multiplied by the odds of winning: (3/57)3 + (26/57)2 + (28/57)(-1) = 9/57 + 54/57 - 58/57 = 60/57 -58/17 = 2/57 = about 0.578 winnings = $0.578 = 0.58 cents
You will win 0.58 cents each time you play this game, on average.
but the mode is on black. You're more likely to lose (28/57)(-1) about 0.49 cents on your first time to play
but if you stick it out, and play enough times, you'll average 0.58 cents a game
Drop down 1 drop 2 = AOB or BOD
Drop3 = angles addition postulate, d definition of complementary angles, addition, property of equality, definition of perpendicular line segments
There is a missing step between step 8 and 9 because the student did not state that m∠BOC + m∠COD = m∠BOD. This statement is true because of the angles addition postulate.
What is angles addition postulate?The Angle Addition Postulate states that if point P lies in the interior of angle ∠RST, then the measure of ∠RST is equal to the sum of the measures of ∠RSP and ∠PST. In other words, for any angle ∠RST with point P in its interior, says that m∠RST = m∠RSP + m∠PST.
The Angle Addition Postulate is used in geometry to find the measures of angles that are formed by two intersecting lines or when a point is located in the interior of an angle.
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Image transcribed:
Select the correct answer from each drop-down menu.
Given: ∠BOC and ∠COD are complementary angles
lines BO intersects lines AD at point O
Prove: ∠AOB ≅∠BOD
Statements--|--Reasons
1. ∠BOC and ∠COD are complementary angles; BO intersects AD at point O--|--1. given
2. m∠BOC+ m∠COD = 90°--|--2. definition of complementary angles
3. substitution property of equality--|--3. m∠BOD 90°
4. ∠AOB and ∠BOD are supplementary angles--|--4. linear pair theorem
5. m∠AOB + m∠BOD = 180°--|--5. definition of supplementary angles
6. m∠AOB+90° = 180°--|--6. substitution property of equality
7. m∠AOB = 90°--|--7. subtraction property of equality
8. m∠AOB = m/BOD--|--8. substitution property of equality
9. ∠AOB∠ BOD--|--9. definition of congruent angles
There is a step missing in the proof. Identify where in the proof there is a missing step and what this step should be.
There is a missing step between _______ (options 8. and 9.,1. and 2., 2 and 3., 4. and 6.)because the student did not state m∠BOC+m∠COD - m∠______ (options AOB or BOD). This statement is true because of the _____ (options angles addition postulate, d definition of complementary angles, addition, property of equality, definition of perpendicular line segments)
Find 4/5 of twelve times a
third of sixty-nine
Answer:
220.8
Step-by-step explanation:
4/5 × (12 × 1/3) × 69
4/5 × (4) × 69
3.2 × 69
= 220.8
Helppp ASAP, what is a conjecture for this
Probability and statistics question
(a) There are 0.1271 percent of poodles that weigh more than 7.9 kilograms. (b) The region to a left , z1 is 0.2389, and the proportion of pups with weights of no more than 7.4 kilos is 0.2852.
How much does a kilogram weigh?a. i. To determine how many times the revenue in 2017 was greater than the revenue in 2016, divide the revenue in 2017 by the revenue in 2016:
$164,900\div 138,200 = 1.1939$
Revenue in 2017 was 1.1939 times greater than revenue in 2016.
ii. To calculate the revenue in 2017 as a percentage of the revenue in 2016, divide the revenue in 2017 by the revenue in 2016 and multiply by 100:
$dfrac164,900138,200 multiplied by 100% = 119.39%$In 2017, revenue was 119.39% higher than in 2016.
b. i. To determine how many times the population of Huntersville was greater than the population of Wright's Park in 2015, divide the population of Huntersville by the population of Wright's Park:
(a) To find the proportion of poodles with weights over 7.9 kilograms, we need to find the area under the standard normal distribution curve to the right of z = (7.9 - 7) / 0.7 = 1.14. From the standard normal distribution table, the area to the right of z = 1.14 is 0.1271. Therefore, the proportion of poodles with weights over 7.9 kilograms is 0.1271.
(b) To find the proportion of poodles with weights of at most 7.4 kilograms, we need to find the area under the standard normal distribution curve to the left of z = (7.4 - 7) / 0.7 = -0.86. From the standard normal distribution table, the area to the left of z = -0.86 is 0.1949. Therefore, the proportion of poodles with weights of at most 7.4 kilograms is 0.1949.
(c) To find the proportion of poodles with weights between 6.5 and 7.8 kilograms inclusive, we need to find the area under the standard normal distribution curve between z = (6.5 - 7) / 0.7 = -0.714 and z = (7.8 - 7) / 0.7 = 1.14. From the standard normal distribution table, the area to the left of z = -0.714 is 0.2389 and the area to the right of z = 1.14 is 0.1271. Therefore, the proportion of poodles with weights between 6.5 and 7.8 kilograms inclusive is 1 - 0.2389 - 0.1271 = 0.634.
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An archway is shown. A semicircle top arch sits on two rectangular pillars. The rectangular pillars are 3 meters wide. The distance between the 2 pillars is 6 meters. The rectangular pillars have a height of 4 meters.
Determine the area of the archway with a semicircle top arch and two rectangular pillars.
The lower supports are
and the area of the two supports is
square meters.
The upper arch can be decomposed as one semicircle with radius
meters minus a semicircle with radius 3 meters.
The area of the archway is (
π + 24) square meters.
The area of the archway is (π + 24) square meters, as the total area of the two rectangular pillars is 24 square meters, area of semicircle is (9π) / 2 square meters.
What is radius?Radius is a term used in geometry to refer to the distance from the center of a circle or sphere to its edge. It is usually represented by the symbol "r". In a circle, all points on the edge, also called the circumference, are at the same distance from the center. This distance is equal to the radius. The radius is half the length of the diameter, which is the distance across the circle passing through its center.
To determine the area of the archway, we need to calculate the area of the two rectangular pillars and the semicircle top arch, and then add them together.
The area of each rectangular pillar is:
Area of each pillar = length x width
= 4 meters x 3 meters
= 12 square meters
So the total area of the two rectangular pillars is:
Total area of pillars = 2 x 12 square meters
= 24 square meters
Now we need to calculate the area of the semicircle top arch. The radius of the semicircle is half of the distance between the two pillars, which is:
Radius of semicircle = (6 meters / 2) = 3 meters
The area of the semicircle is:
Area of semicircle = (π x r²) / 2
= (π x 3²) / 2
= (9π) / 2 square meters
We also need to subtract the area of the rectangle sections at the top of each pillar from the semicircle to get the area of the archway. Each of these rectangles has a length of 3 meters and a height of the difference between the radius of the semicircle and the height of the pillars, which is:
Height of rectangle = (3 meters - 4 meters) = -1 meter
Since we can't have a negative height, we take the absolute value of this difference:
|Height of rectangle| = 1 meter
The area of each rectangle is:
Area of each rectangle = length x width
= 3 meters x 1 meter
= 3 square meters
So the total area of the two rectangles is:
Total area of rectangles = 2 x 3 square meters
= 6 square meters
Therefore, the area of the archway is:
Area of archway = Area of pillars + Area of semicircle - Area of rectangles
= 24 square meters + (9π) / 2 square meters - 6 square meters
= (π + 24) square meters
Hence, the area of the archway is (π + 24) square meters.
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Antiderivative of
(tan^-x× e^tan^-1x)/(1+x^2)
Let u = tan⁻¹x, then du/dx = 1/(1+x²).
Using the formula for the derivative of inverse tangent, we have:
tan(u) = x
sec²(u) du/dx = 1
du/dx = cos²(u)
Substituting into the original expression, we get:
∫(tan⁻¹x × e^tan⁻¹x)/(1+x²) dx = ∫(tan⁻¹x × e^u × cos²(u)) du
Using integration by parts with u = tan⁻¹x and dv = e^u × cos²(u) du, we get:
v = (1/2) e^u (sin(u) + u cos(u))
∫(tan⁻¹x × e^u × cos²(u)) du = (1/2) e^u (sin(u) + u cos(u)) tan⁻¹x - ∫[(sin(u) + u cos(u)) / (1+x²)] dx
Substituting back u = tan⁻¹x, we get:
∫(tan⁻¹x × e^tan⁻¹x × cos²(tan⁻¹x)) dx = (1/2) e^tan⁻¹x (x sin(tan⁻¹x) + cos(tan⁻¹x)) tan⁻¹x - ∫[(x cos(tan⁻¹x) + sin(tan⁻¹x)) / (1+x²)] dx
Using the identity sin(tan⁻¹x) = x / √(1+x²) and cos(tan⁻¹x) = 1 / √(1+x²), we simplify the expression to:
∫(tan⁻¹x × e^tan⁻¹x × cos²(tan⁻¹x)) dx = (1/2) x e^tan⁻¹x + (1/2) ∫[e^tan⁻¹x / (1+x²)] dx
The remaining integral can be solved using another substitution with v = tan⁻¹x, which results in:
∫(tan⁻¹x × e^tan⁻¹x × cos²(tan⁻¹x)) dx = (1/2) x e^tan⁻¹x + (1/2) ln(1+x²) + C, where C is the constant of integration
2. Sundaes come with vanilla or chocolate ice cream. The toppings are hot fudge and
caramel. The final choice is whipped cream or marshmallow.
• What is the probability of getting a sundae with vanilla ice cream, caramel, and
marshmallow?
● Draw a tree diagram to find the solution.
• Justify mathematically that your answer is correct.
Answer:
Step-by-step explanation:
The tree that shows the correct combination of one flavor of ice cream and one topping is the the first tree.
What is tree diagram ?
A tree diagram is a tool in the fields of general mathematics, probability, and statistics that helps calculate the number of possible outcomes of an event or problem, and to cite those potential outcomes in an organized way.
Tree diagrams, also known as probability trees or decision trees, are quite versatile and may be useful in many fields, including finance.
A tree diagram lets a user start at a single point and make mutually exclusive decisions or experience mutually exclusive events to follow a path down the branches of the tree. Using a tree diagram is simple once you assign the appropriate values to each node.
From the question, and the definition it is clear that vanilla splits into fudge and caramel and chocolate to sprinkles.
Hence A is the correct option.
Can someone tell me if I did this correctly? Find the value of x and simplify to radical form and explain if I got it wrong (which I probably did)
Answer:
Step-by-step explanation:
SOHCAHTOA [tex]sin =\frac{opp}{hyp} ,cos=\frac{adj}{hyp} ,tan=\frac{opp}{adj}[/tex]
[tex]sin60=\frac{x}{8}[/tex]
[tex]x=8sin60=8\frac{\sqrt{3} }{2} =4\sqrt{3}[/tex]
please help immediately
The numbers which replaces the capital letters are A is -1, B is 1, C is 3, D is 0, E is 5 and F is -2.
What is logarithm?A logarithm is a mathematical function that measures the relationship between two quantities, by expressing one quantity in terms of another. More specifically, a logarithm is the power to which a base must be raised to produce a given number.
According to question:a) [tex]2^(A) * 2^(B) = 2^0[/tex] can be rewritten as [tex]2^(A+B) = 2^0[/tex].
To find the values of A and B, we need to solve the equation [tex]2^(A+B) = 2^0\\[/tex].
Since [tex]2^0[/tex] = 1, we have [tex]2^(A+B)[/tex] = 1, which means A + B = 0. One possible solution is A = -1 and B = 1, since (-1) + 1 = 0.
Therefore, the equation becomes [tex]2^(-1) * 2^1 = 2^0[/tex], which is true.
(A) = -1
(B) = 1
b) [tex](2^3)/(2^(C)) = 2^(D)[/tex] can be simplified as [tex]2^(3-C) = 2^(D)[/tex].
To find the values of C and D, we need to set the exponents equal to each other. So, we have 3 - C = D, which means C = 3 - D.
One possible solution is C = 3 and D = 0, since 3 - 0 = 3.
Therefore, the equation becomes [tex](2^3)/(2^3) = 2^0[/tex], which is true.
(C) = 3
(D) = 0
c) [tex]2^(-3) * (E)^(-3) = 10^(F)[/tex] can be rewritten as [tex](1/2^3) * (1/E^3) = 10^F[/tex]. Multiplying both sides by [tex]2^3 * E^3[/tex], we get [tex]1 = 10^F * 2^3 * E^3[/tex].
Taking the logarithm of both sides, we get log(1) = [tex]log(10^F * 2^3 * E^3)[/tex]. Since log(1) = 0, we have 0 = F log(10) + 3log(2) + 3log(E), which simplifies to F = -2 - 3log(E)/log(10) with one possible solution of E = 5 and F = -2. Therefore, the equation becomes [tex]2^(-3) * 5^(-3) = 10^(-2)[/tex], which is true.
(E) = 5
(F) = -2
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