We have the following system of equations:
x₁ + 0.5x₂ + 0.1x₃ = 100
0.25x₁ + x₂ + 0.2x₃ = 75
0.3x₁ + 0.4x₂ + x₃ = 200
Define the term equation?Statement gives the equality of different mathematical expressions is called as an equation.
Let x₁, x₂, and x₃ be the outputs of I₁, I₂, and I₃, respectively. Then the total production must satisfy the following system of equations:
For I₁:
I₁ itself produces x₁ unitsIt requires 0.5 units of I₂ for every unit, so it requires 0.5x₂ unitsIt requires 0.1 units of I₃ for every unit, so it requires 0.1x₃ unitsTherefore, total demand from I₁ is x₁ + 0.5x₂ + 0.1x₃ = 100For I₂:
I₂ itself produces x₂ unitsIt requires 0.25 units of I₁ for every unit, so it requires 0.25x₁ unitsIt requires 0.2 units of I₃ for every unit , so it requires 0.2x₃ unitsTherefore, total demand from I₂ is x₂ + 0.25x₁ + 0.2x₃ = 75For I3:
I₃ itself produces x₃ unitsIt requires 0.3 units of I₁ for every unit, so it requires 0.3x₁ unitsIt requires 0.4 units of I₂ for every unit, so it requires 0.4x₂ unitsTherefore, total demand from I₃ is x₃ + 0.3x₁ + 0.4x₂ = 200Therefore, the system equations are:
x₁ + 0.5x₂ + 0.1x₃ = 100
0.25x₁ + x₂ + 0.2x₃ = 75
0.3x₁ + 0.4x₂ + x₃ = 200
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Which of the following uses the distributive property correctly?
3(x+5)=3-x+5
3(x+5)=3-x-3-5
3(x+5)=3+3.5
3(x+5)=3-x+3.5
Answer:
None of them, 3x + 15 is correct.
Step-by-step explanation:
In order to apply the distributive properly correctively, we need to multiply everything in the parentheses. For an example, lets look at the given expression:
[tex]3(x+5)[/tex]
We multiply everything in the parentheses by 3
[tex]3x+15[/tex]
Looking at out choices, none of them uses the distributive property correctly.
HELP! The average high temperatures in degrees for a city are listed.
58, 61, 71, 77, 91, 100, 105, 102, 95, 82, 66, 57
If a value of 110.5° is added to the data, how does the mean change and by how much?
The mean increases by 2.3°.
The mean increases by 2.7°.
The means stays at 80.4°.
The mean stays at 82.7°.
Which equation represents the hanger balance?
hurry!!!!
The correct equation for the hanger balance is:= w = 2
How to balance equations?
The hanger balance equation is typically used in physics experiments to determine the weight of an object by comparing it to known weights on the other side of a balance. The equation is based on the principle of equilibrium, which states that the sum of the forces acting on an object must be zero for it to remain stationary.
The correct equation for the hanger balance is:
w = 2
This equation simply states that the weight of the object being measured (represented by "w") is equal to a known weight of 2 units. This indicates that the object being measured has a weight of 2 units.
The other equations provided, such as "6 = w + 2" and "6 = w - 2", do not accurately represent the hanger balance equation because they do not take into account the known weight on the other side of the balance. In order to determine the weight of an object using a hanger balance, it is necessary to have a known weight on the other side of the balance to compare it to.
In summary, the correct equation for the hanger balance is w = 2, which represents the weight of the object being measured as 2 units.
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A triangle has angles that measure 55°, 80°, and r. What is r?
Answer: r = 45°
Step-by-step explanation:
All triangles angles add up to 180 degrees.
With that said we can use the given information and create an equation to solve.
55° + 80° + r = 180°
Lets solve for r.
55° + 80° + r = 180°
first lets combine like terms.
135° + r = 180°
now lets subtract 143° from both sides
135° + r = 180°
-135° -135°
r = 45°
The numbers 2 through 16 are written on tiles and put into a hat. One number is drawn at random. Determine the theoretical probability of drawing a prime number.
Answer:
2/5 or 0.4, which is also equivalent to a 40% chance.
Step-by-step explanation:
To determine the theoretical probability of drawing a prime number from the numbers 2 through 16, we need to first identify the prime numbers in this range. The prime numbers between 2 and 16 are 2, 3, 5, 7, 11, and 13.
There are a total of 15 numbers in the hat, and 6 of them are prime. Therefore, the probability of drawing a prime number can be calculated as:
Probability of drawing a prime number = Number of favorable outcomes / Total number of possible outcomes
Number of favorable outcomes = 6 (since there are 6 prime numbers in the range of 2 through 16)
Total number of possible outcomes = 15 (since there are 15 numbers in the hat)
So, the probability of drawing a prime number is:
Probability of drawing a prime number = 6 / 15
Simplifying this fraction by dividing both numerator and denominator by 3, we get:
Probability of drawing a prime number = 2 / 5
Therefore, the theoretical probability of drawing a prime number from the numbers 2 through 16 is 2/5 or 0.4, which is also equivalent to a 40% chance.
when a confounding variable is present in an experiment, one cannot tell whether the results were due to the a) independent variable or the dependent variable. b) independent variable or the confounding variable. c) dependent variable or the interval variable. d) dependent variable or the participant variable.
When a confounding variable is present in an experiment, one cannot tell whether the results were due to the independent variable or the dependent variable. The correct option is a) independent variable or the dependent variable.
A confounding variable is any variable that influences the dependent variable or response variable and is also
correlated with the independent variable. Because it is difficult to determine which variable is causing changes in the
dependent variable, it is also known as a confounding variable.
In other words, a confounding variable is a variable that can affect the dependent variable in an experiment, making it
impossible to determine the relationship between the independent variable and the dependent variable.
Therefore, when a confounding variable is present in an experiment, one cannot tell whether the results were due to
the independent variable or the dependent variable.
Consequently, researchers must take steps to identify and eliminate or control the effect of confounding variables to obtain valid and reliable results.
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in a binomial probability problem, if the nature of the problem is such that it is feasible to use either the binomial tables or the normal distribution to approximate the probability, which would give the more accurate answer?
If the sample size is large and the probability of success is not too close to 0 or 1, using the normal distribution to approximate the binomial probability will generally give a more accurate answer than using the binomial tables. However, if the sample size is small or the probability of success is very close to 0 or 1, using the binomial tables may be more accurate.
If the sample size is large (typically, n ≥ 30) and suppose that the probability of success (p) is not too close to 0 or 1, then we can use the normal distribution. As it gives more approximate value to the binomial probability which will generally give a more accurate answer than using the binomial tables.
This is because the normal distribution provides a more precise approximation of the binomial distribution as the sample size increases.
However, if the the probability of success is very close to 0 or 1 or sample size is small then using the binomial tables to calculate the probability directly may be more accurate. In general, it's important to check the conditions for using the normal approximation before using it and to use good judgment in deciding which method is more appropriate for a given problem.
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A new line is drawn that passes through the point (-6,2) and is parallel to line m.
The equation of this line can be written as y=1/2x+a Enter the value of a.
60000 dividido en 800
Answer:
75
Step-by-step explanation:
Fill in the table for side length and area of different squares
a) The area of the square for side length [tex]3[/tex] is [tex]9[/tex][tex]cm^{2}[/tex].
The area of the square for side length [tex]100[/tex] is [tex]10000[/tex][tex]cm^{2}[/tex]
The area of the square for side length [tex]25[/tex] is [tex]625cm^{2}[/tex].
The area of the square for the side length [tex]s[/tex] is [tex]s^{2}cm^{2}[/tex].
b) No the side length of the square and area are not directly proportional.
What is the area of a square?The area of a square is the amount of space enclosed within the square, and it can be calculated using the formula A = [tex]s^{2}[/tex], where A is the area and s is the length of one side of the square.
According to the given information
The area of a square is the amount of space enclosed within the square, and it can be calculated using the formula A =[tex]s^{2}[/tex]
[tex]A = s^{2} = 3^{2} = 9cm^{2}[/tex]
[tex]A = s^{2} = 10^{2} = 10000cm^{2}[/tex]
[tex]A = s^{2} = 25^{2} = 625cm^{2}[/tex]
[tex]A = s^{2} = s^{2} = s^{2} cm^{2}[/tex]
Therefore area of the square are [tex]9cm^{2} ,10000cm^{2}, 625cm^{2} ,s^{2}cm^{2}[/tex]
Therefore the side length of the square and area are not directly proportional they are proportional to the square root of area.
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Graph g(x), where f(x) = 2x − 5 and g(x) = f(x + 1). a line labeled g of x that passes through points 0, negative 4 and 2, 0 a line labeled g of x that passes through points 0, negative 3 and 4, 5 a line labeled g of x that passes through points negative 5, negative 3 and 0, 7 a line labeled g of x that passes through points 0, negative 7 and 5, 3
To graph g(x), we must replace x with x + 1 in the function f(x) = 2x - 5: g(x) = f(x + 1) = 2(x + 1) - 5 = 2x - 3 (a) To display the g of x line that passes through the coordinates 0, -4, and 2, 0, we replace x = 0 and x = 2 in the equation g(x) = 2x - 3:
g(0) = 2(0) - 3 = -3
g(2) = 2(2) - 3 = 1
Hence the line's points are (0, -3) and (2, 1):
yaml Copy code | 5 |- * (2,1) | 4 |- | 3 |- * (0,-3) | 2 |- | 1 |- | 0 |—————————— 0 1 2 3 4 5\s(b) (b) To display the g of x line that passes through points 0, -3, 4, 5, we replace x = 0 and x = 4 in the equation g(x) = 2x. - 3:
g(0) = 2(0) - 3 = -3
g(4) = 2(4) - 3 = 5
Hence the line's points are (0, -3) and (4, 5):
Copy code in yaml | 5 |- * (4,5)
|\s4 |-\s |\s3 |- * (0,-3)\s |\s2 |-\s |\s1 |-\s |\s0 |
————————-\s 0 1 2 3 4 5\s(c) (c) To display the line labeled g of x that goes through points -5, -3, and 0, 7, we use the equation g(x) = 2x - 3:
g(-5) = 2(-5) (-5) - 3 = -13\sg(0) = 2(0) - 3 = -3
Thus the line points are (-5, -13) and (0, -3):
| 5 |- | 4 |- | 3 |- | 2 |- | 1 |- | 0 |- * (0,-3) | -1 |- (-5,-13)
\ \s ——————-\s -5 -4 -3 -2 -1 0\s(d) To draw the g of x line that passes between points 0, -7, and 5, 3, In the equation g(x) = 2x - 3 we substitute x = 0 and x = 5:
g(0) = 2(0) - 3 = -3
g(5) = 2(5) (5) - 3 = 7
Hence the line's points are (0, -3) and (5, 7):
| 7 |- * yaml Copy code (5,7)
|\s6 |-\s |\s5 |- \s | * (0,-3) (0,-3)
4 |-\s |\s3 |-\s |\s2 |-\s |\s1 |-\s |\s0 |
————————-\s 0 1 2 3 4 5.
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Question 7(Multiple Choice Worth 2 points)
(Scatter Plots MC)
A marine biologist was interested in seeing the relationship between a manatee's weight and food consumption. She gathered data and wanted to construct a graph to display the data in order to observe this relationship.
Is a scatter plot or a line graph more appropriate to construct for the data, given the context of the problem? Explain.
A line graph, because it would show the association or relationship between two variables
A line graph, because it would show how a variable changes over time between each data point given
A scatter plot, because it would show the association or relationship between two variables
A scatter plot, because it would show how a variable changes over time between each data point given
A scatter plot would be more appropriate to construct for the data, given the context of the problem. The reason is that the marine biologist is interested in seeing the relationship between a manatee's weight and food consumption.
A scatter plot is a type of graph that displays the relationship between two variables, where each variable is plotted along one axis, and a point is placed at the intersection of the two values. In this case, weight would be plotted along one axis and food consumption along the other axis.
A scatter plot allows the researcher to see if there is a relationship between the two variables, and if so, what type of relationship it is (positive, negative, or no correlation). Therefore, a scatter plot is the most appropriate way to display the relationship between a manatee's weight and food consumption.
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a paperback book company charges $6 for each paperback book plus $4 for shipping and handling per order. what would the total cost for 3 books.
ANSWER FAST
Answer:
$30
Step-by-step explanation:
6x3=18
+
4x3=12
=30
Inga is solving 2x2 + 12x – 3 = 0. Which steps could she use to solve the quadratic equation? Select three options. 2(x2 + 6x + 9) = 3 + 18 2(x2 + 6x) = –3 2(x2 + 6x) = 3 x + 3 = Plus or minus StartRoot StartFraction 21 Over 2 EndFraction EndRoot 2(x2 + 6x + 9) = –3 + 9
She could solve for (x+3)² and find two solutions, x = (-3 + √(33/2)) / 2 and x = (-3 - √(33/2)) / 2.
What is quadratic equation?it's a second-degree quadratic equation which is an algebraic equation in x. Ax² + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term, is the quadratic equation in its standard form.
The three steps that Inga could use to solve the quadratic equation 2x² + 12x – 3 = 0 are:
1. Use the quadratic formula: Inga could use the quadratic formula x = (-b ± √(b² - 4ac)) / 2a, where a = 2, b = 12, and c = -3.
2. Factor the quadratic expression: Inga could factor the quadratic expression into (2x - 1)(x + 3) = 0 and then use the zero product property to find the solutions x = 1/2 and x = -3.
3. Complete the square: Inga could use the method of completing the square by adding and subtracting (6/2)² = 9 to the left side of the equation, giving 2(x+3)² - 33 = 0. Then she could solve for (x+3)² and find two solutions, x = (-3 + √(33/2)) / 2 and x = (-3 - √(33/2)) / 2.
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will give brainlest
|x+3|=9
you have to spilt the equation to get the absolute value
Answer:
To solve the equation |x+3|=9, we need to split it into two separate equations, one for the positive value and one for the negative value of the absolute value:
For the positive value:
x+3=9
Solving for x, we get x=6
For the negative value:
-(x+3)=9
Multiplying both sides by -1, we get:
x+3=-9
Solving for x, we get x=-12
Therefore, the solutions to the equation |x+3|=9 are x=6 and x=-12.
Hope This Helps!
the infographic shows percentages of women overall in elected office as well as the share of latinas in elected office. women constitute approximately what percentage of members of congress, statewide executives, state legislators, and mayors in the top 100 cities? check all that apply.
According to the infographic, women constitute approximately 15-19% of members of Congress, less than 15% of state legislators, and 25-50% of statewide executives and mayors in the top 100 cities.
According to the information provided in the infographic, the following options apply:
Women constitute approximately 25-50% of statewide executives and mayors in the top 100 cities.
Women constitute approximately 15-19% of members of Congress.
Women constitute less than 15% of state legislators.
Therefore, the correct options are:
15-19%
Less than 15%
25-50%
The infographic presents data on the percentage of women overall in elected office and the share of Latinas in elected office. The data indicates that women constitute less than 15% of state legislators, approximately 15-19% of members of Congress, and 25-50% of statewide executives and mayors in the top 100 cities. This suggests that women are underrepresented in elected office at all levels of government, with the exception of statewide executives and mayors in the top 100 cities. The infographic underscores the need for greater gender diversity in elected office, and the importance of supporting and encouraging women to run for political office at all levels of government.
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Complete question:
The infographic shows percentages of women overall in elected office as well as the share of Latinas in elected office. Women constitute approximately what percentage of members of Congress, statewide executives, state legislators, and mayors in the top 100 cities? Check all that apply.
19-25%
Less than 15%
15-19%
25-50%
Emily bought a car that depreciates at a
rate of 18% per year. She paid $55,000 for
the car. How much will her car be worth
after 8 years.
After answering the presented question, we can conclude that As a equation result, Emily's automobile will be valued roughly $19,105.25 after 8 years.
What is equation?An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "[tex]2x + 3 = 9[/tex]" asserts that the phrase "[tex]2x + 3[/tex]" equals the value "9." The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. The variable x is raised to the second power in the equation "[tex]x2 + 2x - 3 = 0[/tex]." Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.
Emily's car will be worth 100% - 18% = 82% of its original value after one year, or 0.82 times its initial value.
After two years, the car is worth 0.82 times its initial value, or 0.822% of its original value.
Similarly, the car will be worth 0.828% of its initial value after 8 years.
Hence, after 8 years, Emily's automobile will be worth:
In 8 years, the value is $55,000 x 0.82.
After 8 years, the value is $19,105.25. (rounded to the nearest cent)
As a result, Emily's automobile will be valued roughly $19,105.25 after 8 years.
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Given � ( � ) = − 2 � 2 − 4 � + 13 f(x)=−2x 2 −4x+13, find � ( − 7 ) f(−7)
The function g(-7) * f(-7) is approximately equal to *-8056.5*.
What is a function?
A function is a mathematical rule that assigns a unique output value to each input value from a given set. It represents a relationship between two sets of numbers in which each input has exactly one output. A function can be expressed symbolically, graphically, or verbally and is used to model real-world phenomena, solve equations, and make predictions in various fields of study.
According to the given information
To find g(-7) and f(-7), we need to substitute -7 for y in the function g(y) = -2ᵇ - 4b + 13 and x in the function f(x) = -2x² - 4x + 13.
g(-7) = -2⁻⁷ - 4⁻⁷ + 13
Simplifying this expression gives:
g(-7) = -1/128 + 28 + 13
g(-7) = 141 - 1/128
Therefore, g(-7) is approximately equal to 140.9922 .
f(-7) = -2(-7)² - 4(-7) + 13
Simplifying this expression gives:
f(-7) = -98 + 28 + 13
f(-7) = -57
Therefore, f(-7) is equal to -57.
So, g(-7) * f(-7) is approximately equal to -8056.5.
The correct question is
g(y ) = − 2ᵇ − 4b + 13 and f(x)=−2x² −4x+13, find g( − 7 ) f(−7)
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make absolute value equation of
0 and 8
5 and 17
-4 and 6
in the form |x-c|=d
Please hurry!!!!
Answer:
Sure, here are the absolute value equations of 0 and 8, 5 and 17, and -4 and 6 in the form $|x-c|=d$:
* For 0 and 8, $|x-0|=8$. This means that $x-0=8$ or $x-0=-8$. Solving for $x$, we get $x=8$ or $x=-8$.
* For 5 and 17, $|x-5|=17$. This means that $x-5=17$ or $x-5=-17$. Solving for $x$, we get $x=22$ or $x=-12$.
* For -4 and 6, $|x-(-4)|=6$. This means that $x-(-4)=6$ or $x-(-4)=-6$. Solving for $x$, we get $x=10$ or $x=-2$.
I hope this helps! Let me know if you have any other questions.
Step-by-step explanation:
the number 57 belongs to which of the following sets of the numbers? N only, N, W and Z only, N, W, Z, and Q only, All of the following: N, W, Z, Q, and R
The number 57 belongs to the sets Z, Q, and R, but not N or W.
What are natural numbers ?
Natural numbers are the counting numbers that we use to represent quantities and perform arithmetic operations. These are the numbers that we use to count objects, such as apples, books, or people.
The number 57 belongs to the following sets of numbers:
N (the natural numbers): No, because 57 is not a counting number.
W (the whole numbers): No, because 57 is not a non-negative integer.
Z (the integers): Yes, because 57 is an integer (a positive integer, in fact).
Q (the rational numbers): Yes, because 57 can be written as a ratio of two integers (57/1).
R (the real numbers): Yes, because every rational number is also a real number.
Therefore, the number 57 belongs to the sets Z, Q, and R, but not N or W.
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Complete question -
10 to the 9th power is how many times 10 to the 8th power?
10 to the 9th power is 10 times 10 to the 8th power.
NEED HELP ASAP PLEASE.....
The output of the function g(x) = [tex]3^{(\frac{x}{2} )}[/tex] when x = 2 is 3.
How to solve function?g(x) is a function as defined as follows:
g(x) = [tex]3^{(\frac{x}{2} )}[/tex]
If we input n2 into the g(x), the output g(x) can be gotten as follows:
Therefore,
g(x) = [tex]3^{(\frac{x}{2} )}[/tex]
where
x = 2
Therefore,
[tex]3^{(\frac{2}{2} )}[/tex] = 3¹
Therefore,
the output is 3.
g(2) = 3
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any one know the answers?
The value of the boxes, A , B and C are 22/4, 15/4 and 9 1/4 respectively
How to determine the valuesNote that from the information given, we have that the number in a box is the sum of the two numbers below it.
Then, we can deduce that;
A = The sum of the two mixed fractions, 3 1/4 and 2 1/4
B = The sum of the two mixed fractions, 2 1/4 and 1 1/2
Now, add the values
A = 3 1/4 + 2 1/4
convert to improper fractions
A = 13/4 + 9/4
Add the values
A = 22/4
B = 2 1/4 + 1 1/2
convert to improper fractions
B = 9/4 + 3/2
B =9 + 6 /4
B = 15/4
Then , the value of C = A + B
C = 22/4 + 15/4
C = 37/4
C = 9 1/4
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water is leaking out of an inverted conical tank at a rate of 10,500 cm3/min at the same time that water is being pumped into the tank at a constant rate. the tank has height 6 m and the diameter at the top is 4 m. if the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate (in cm3/min) at which water is being pumped into the tank.
The rate at which water is being pumped into the tank is 11,760 cm3/min.
How to find rate?The formula for the volume of a conical tank is:V = (1/3)πr2h
Where r is the radius of the tank, and h is the height of the tank.Find the radius of the tank at a height of 2 m.Using similar triangles:
(r / 2) = (2 / 6)
r = 2/3 * 4r = 8/3 cm
The formula for the rate of change of volume of a conical tank is:dV / dt = (πr2 / 3)dh / dt
dV / dt = pump rate - leak rate
= pump rate - 10,500 cm3/mindh / dt
= 20 cm/minr = 8/3 cm
Plug in the values:pump rate - 10,500 = (π(8/3)2 / 3) * 20pump rate - 10,500
= 2114.67pump rate
= 11,760 cm3/min
Therefore, the rate at which water is being pumped into the tank is 11,760 cm3/min.
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help me pls!! the assignment is due by midnight
Answer: 1. Chord 2. Diameter
Answer:
Step-by-step explanation:
Since A and G are on a line thats not in the middle of the circle, the clear answer would be a chord
Since C is a point on the middle of the circle and B is at the end of the circle, the answer would be radius
PLEASE HELP ILL GIVE BRAINLIEST
Answer:
True
Step-by-step explanation:
Since C has to add up to 180 and we already have 150 we can assume that that other angle is 30. The sum of angles in a triangle always has to equal 180. 120+30=150 meaning B also equals 30. B+A=150 therefore making it true.
A pole leans away from the sun at an angle of 7° to the vertical. When the elevation of the sun is 44°, the
pole casts a shadow 49 ft long on level ground. How long is the pole?
The length of the pole is approximately 412.7 feet.
What is the tangent function?
The tangent function is a mathematical function that relates the ratio of the length of the side opposite to an acute angle in a right triangle to the length of the adjacent side. It is defined as the ratio of the sine of the angle to the cosine of the angle.
We can solve this problem using trigonometry. Let's call the length of the pole "x". We can then use the tangent function to relate the angle that the pole makes with the vertical to the length of the shadow cast by the pole.
The tangent of the angle between the pole and the vertical is equal to the opposite side (length of the shadow) divided by the adjacent side (length of the pole).
tan(7°) = opposite / adjacent
tan(7°) = 49 / x
We can solve for x by multiplying both sides by x and then dividing both sides by tan(7°):
x = 49 / tan(7°)
Now we know the length of the pole is:
x = 49 / tan(7°) = 412.7 ft (rounded to one decimal place)
Therefore, the length of the pole is approximately 412.7 feet.
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Anyone help in number1 (d)
What are the values of angle a c and f
Answer:
Step-by-step explanation:
f=133
a=93
c=140
Pilar is playing with a remote-controlled toy boat. She puts the boat in a lake and it travels 400m at a constant speed. On the way back to Pilar, the boat travels the same route at the same speed for 2 minutes, and then Pilar uses the remote control to increase the boat's speed by 10 m/min. So the return trip is 60 seconds faster. How long does the return trip take?
Answer:
Therefore, the time for the return trip is:
≈ 5.306 seconds
Step-by-step explanation:
Let's start by finding the speed of the boat. We know that the boat travels 400m at a constant speed, so we can use the formula:
Speed = Distance / Time
Speed = 400m / Time (for one-way trip)
Let's assume that the time for the one-way trip is t. Then we can rewrite the speed formula as:
Speed = 400m / t
Now let's find the time for the return trip. We know that the boat travels the same route at the same speed for 2 minutes, which is equivalent to 120 seconds. This means that the distance traveled on the way back is:
Distance = Speed x Time
Distance = (400m / t) x 120 seconds
Distance = 48000m / t
We also know that when Pilar increases the boat's speed by 10 m/min, the return trip is 60 seconds faster. This means that the time for the return trip is:
t - 60 seconds
Using the same formula as before, the distance traveled on the return trip is:
Distance = (400m / (t + 1/2)) x (t - 60) seconds
Distance = (400m / (t + 1/2)) x (t - 60/60) minutes
Distance = 400m x (t - 60/60) / (t + 1/2) minutes
Now we know that the distance traveled on the way back is equal to the distance traveled on the way there:
48000m / t = 400m x (t - 60/60) / (t + 1/2)
Simplifying this equation, we get:
48 / t = 4 x (t - 1/3) / (t + 1/2)
Multiplying both sides by (t + 1/2), we get:
48(t + 1/2) / t = 4(t - 1/3)
Simplifying this equation, we get:
96 / t = 4t - 4/3
Multiplying both sides by t, we get:
96 = 4t^2 - 4/3t
Multiplying both sides by 3, we get:
288 = 12t^2 - 4t
Rearranging this equation, we get
12t^2 - 4t - 288 = 0
Dividing both sides by 4, we get:
3t^2 - t - 72 = 0
Using the quadratic formula, we can solve for t:
t = [1 ± sqrt(1 + 4(3)(72))] / 6
t = [1 ± sqrt(865)] / 6
Since t must be positive, we take the positive root:
t ≈ 6.366
Therefore, the time for the return trip is:
t - 60 seconds ≈ 6.366 - 60 seconds ≈ 5.306 seconds