The maximum rate of change occurs in the direction of this unit vector.
(a) To find the maximum rate of change of f at point P(1,0), we need to find the gradient of f at that point and then find its magnitude. The direction of maximum increase is given by the unit vector in the direction of the gradient.
The gradient of f is:
∇f(x,y) = <y cos(xy), x cos(xy)>
At point P(1,0), we have:
∇f(1,0) = <0, cos(0)> = <0, 1>
The magnitude of the gradient is:
||∇f(1,0)|| = sqrt([tex]0^2[/tex] +[tex]1^2[/tex]) = 1
Therefore, the maximum rate of change of f at point P is 1, and it occurs in the direction of the unit vector in the direction of the gradient:
u = <0, 1>/1 = <0, 1>
So the maximum rate of change occurs in the y-direction.
(b) To find the maximum rate of change of f at point P(8,1.3), we need to find the gradient of f at that point and then find its magnitude. The direction of maximum increase is given by the unit vector in the direction of the gradient.
The gradient of f is:
∇f(x,y,z) = <2x, 2y, 2z>
At point P(8,1.3), we have:
∇f(8,1.3) = <16, 2.6, 2(1.3)> = <16, 2.6, 2.6>
The magnitude of the gradient is:
||∇f(8,1.3)|| = sqrt[tex](16^2 + 2.6^2 + 2.6^2)[/tex]= sqrt(275.56) ≈ 16.6
Therefore, the maximum rate of change of f at point P is approximately 16.6, and it occurs in the direction of the unit vector in the direction of the gradient:
u = <16, 2.6, 2.6>/16.6 ≈ <0.963, 0.157, 0.157>
So the maximum rate of change occurs in the direction of this unit vector.
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plss helpppssss
6 th grade math
Answer:
Step-by-step explanation:
By the figure, it would mean:
67, 67, 68
72, 72, 73, 76, 76, 77, 78
80, 81, 83, 83, 85, 85, 85, 87, 88
91, 91, 93, 95, 99
a) 2 students
b) 9 students
c) 2 students
Explanation : 5 students (90s) - 3 students (60s) = 2 students
d) 81
Explanation : (67 + 67 + 68 + 72 + 72 + 73 + 76 + 76 + 77 + 78 + 80 + 81 + 83 + 83 + 85 + 85 + 85 + 87 + 88 + 91 + 91 + 93 + 95 + 99) ÷ 24 = 81.33
How many people can fit in the passenger car of 70 feet and a width of 9 feet
Approximately 378 people can fit in a passenger car with a length of 70 feet and a width of 9 feet.
What is the volume of a car with a length of 70 feet and a width of 9 feet?To determine the number of people who can fit in a passenger car with a length of 70 feet and a width of 9 feet, we need to consider the available space inside the car and the amount of space required per person.
Assuming that the passenger car is a rectangular box shape, the volume of the car can be calculated as follows:
Volume of car = Length x Width x Height
Since we are only interested in the number of people who can fit in the car, we will assume a standard height of 6 feet for simplicity. Therefore, the volume of the car can be calculated as follows:
Volume of car = 70 feet x 9 feet x 6 feet
Volume of car = 3,780 cubic feet
Next, we need to determine the amount of space required per person. This can vary depending on the seating arrangement and the size of the individuals, but as a general rule, we can estimate that each person requires approximately 10 to 12 cubic feet of space in a seated position.
Assuming we use the lower end of that estimate and allocate 10 cubic feet of space per person, we can calculate the number of people who can fit in the car as follows:
Number of people = Volume of car / Space per person
Number of people = 3,780 cubic feet / 10 cubic feet per person
Number of people = 378 people
Therefore, if we assume a standard height of 6 feet and allocate 10 cubic feet of space per person, approximately 378 people can fit in a passenger car with a length of 70 feet and a width of 9 feet. However, it is important to note that this is just an estimate and the actual number of people who can fit in the car will depend on factors such as the seating arrangement and the size of the individuals.
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To celebrate halloween, lacey's class is making candy necklaces. lacey is helping pass out string from a 50-yard-spool. she gives 30 inches of string to each student. if there are 24 students in her class, how many yards of string will be leftover?
The class will use 20 yards of the 50-yard spool, leaving 30 yards of string leftover.
This leftover string could be used for future projects or saved for another occasion.
Lacey's class will use a total of 720 inches (30 inches per student x 24 students) of string for the candy necklaces.
To convert this to yards, we divide by 36 (since there are 36 inches in a yard). 720 inches ÷ 36 = 20 yards
It's important to note that when working with different units of measurement, it's necessary to convert them to the same unit before performing calculations.
In this case, we converted inches to yards in order to determine the amount of string used by the class. By doing so, we were able to determine how much string was leftover in yards, which is a more appropriate unit of measurement for a spool of string.
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There is a line through the origin that divides the region bounded by the parabola y = 2x − 7 x^2 and the x-axis into two regions with equal area. What is the slope of that line?
The slope of the line that divides the region into two equal parts is 8/7.
How to find the slope of that line?We begin by finding the x-coordinates of the points where the parabola intersects the x-axis. Setting y = 0, we get:
[tex]2x - 7x^2 = 0[/tex]
x(2 − 7x) = 0
x = 0 or x = 2/7
Thus, the parabola intersects the x-axis at x = 0 and x = 2/7.
We want to find the slope of the line through the origin that divides the region bounded by the parabola and the x-axis into two regions with equal area.
Let's call this slope m.
We know that the area under the parabola from x = 0 to x = 2/7 is:
A = ∫[0,2/7] (2x − 7[tex]x^2[/tex]) dx
A = [[tex]x^2[/tex] − (7/3)[tex]x^3[/tex]] from 0 to 2/7
A = (4/21)
Since we want the line to divide this area into two equal parts, the area to the left of the line must be (2/21).
Let's call the x-intercept of the line h. Then the equation of the line is y = mx, and the area to the left of the line is:
(1/2)h(mx) = (1/2)mhx
We want this to be equal to (2/21), so we can solve for h:
(1/2)mhx = (2/21)
h = (4/21m)
The x-coordinate of the point of intersection of the line and the parabola is given by:
2x − 7[tex]x^2[/tex] = mx
Simplifying, we get:
[tex]7x^2 - (2 + m)x = 0[/tex]
Using the quadratic formula, we get:
[tex]x = [(2 + m) \pm \sqrt((2 + m)^2 - 4(7)(0))]/(2(7))[/tex]
x = [(2 + m) ± √(4 + 4m + [tex]m^2[/tex])]/14
x = [(2 + m) ± (2 + m)]/14
x = 1/7 or x = −(2/7)
Since we want the line to pass through the origin, we must choose x = 1/7, and we can solve for m:
[tex]2(1/7) - 7(1/7)^2 = m(1/7)[/tex]
m = 8/7
Therefore, the slope of the line that divides the region into two equal parts is 8/7.
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Jonana has a board thats 6 ft long she wants to cut it into pieces that are each 1/4 foot long. Write an equation to represent the number of pieces she cut.
Jonana cut 24 pieces of 1/4 foot length from the 6-foot board
How to Write an equation to represent the number of pieces she cutLet "x" be the number of pieces that Jonana cut.
Each piece is 1/4 foot long.
So, the total length of all the pieces is x * 1/4 = x/4 feet.
But the total length is also 6 feet.
So we can set up the equation:
x/4 = 6
Solving for x:
x = 24
Therefore, Jonana cut 24 pieces of 1/4 foot length from the 6-foot board.
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A woman bought 130kg of tomatoes for 52. 0. She sold half of them at a profit of 30%. The rest of the tomatoes started to go bad. She then reduced the selling price per kg by 12%. Calculate
i. The new selling price per kg
ii. The percentage profit on the whole transaction if she threw away 5kg of bad tomatoes
(I) The new selling price per kilogram of the tomatoes is 0.4576.
(II) The percentage profit on the whole transaction is 24.77% if she threw away 5kg of bad tomatoes.
What is the new selling price?The new selling price is calculated as follows;
The cost per kilogram of the tomatoes is;
Cost per kg = Total cost / Total weight
Cost per kg = 52 / 130
Cost per kg = 0.4
Selling price per kg = Cost per kg + (Profit percentage x Cost per kg)
Selling price per kg = 0.4 + (0.3 x 0.4)
Selling price per kg = 0.52
The new selling price per kilogram is:
= Selling price per kg - (Reduction percentage x Selling price per kg)
= 0.52 - (0.12 x 0.52)
= 0.4576
The total revenue from selling the tomatoes is calculated as;
The woman sold half of the 130kg of tomatoes, = 130 / 2 = 65kg
Revenue = (amount sold x selling price per kg) + (amount left x new selling price per kg)
Revenue = (65 x 0.52) + (65 x 0.4576)
Revenue = 33.8 + 29.68
Revenue = 63.48
New total cost = Total cost / Total weight x (Total weight - Bad tomatoes)
New total cost = 0.4 x (130 - 5)
New total cost = 50.6
The profit on the whole transaction is calculated as;
Profit = Total revenue - New total cost
Profit = 63.48 - 50.6
Profit = 12.88
The profit percentage on the whole transaction is calculated as;
Profit percentage = (Profit / Total cost) x 100%
Profit percentage = (12.88 / 52) x 100%
Profit percentage = 24.77%
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Find the surface area of this regular pyramid
The surface area be 243 square feet.
Hence option (d) is correct.
In the given regular pyramid
Slant height = l = 6 ft
Edge of base = s = 9ft
Then,
Area of base = a = 9x9 = 81 square ft
Perimeter of base = p = 4x9 = 36 ft
Since surface area of regular pyramid = A = a + (1/2)ps
= 81 + (36x9)/2
= 81 + 162
= 243
Hence, A = 243 square ft.
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Answer:
The surface area be 243 square feet.
Step-by-step explanation:
The real number properties can be used to simplify numerical expressions. in this section, you will identify which properties were used to simplify several expressions.
which properties were used to simplify the following expression? select all that apply.
4 + 3(9 + 2)
4 + (3 × 9) + (3 × 2)
4 + 27 + 6
4 + 6 + 27
10 + 27
37
The property was not explicitly used in this example, but it is worth noting that adding 0 to any number leaves it unchanged (i.e., a + 0 = a).
How many properties are used to simplify the expression 4 + 3(9 + 2) into 37?The properties that were used to simplify the expression 4 + 3(9 + 2) into 37 are:
Distributive property: The expression was rewritten as 4 + (3 × 9) + (3 × 2) by distributing the 3 over the parentheses.
Associative property: The order of the terms (3 × 9) and (3 × 2) was rearranged without changing the result because of the associative property of multiplication.
Commutative property: The order of the terms 4, 27, and 6 was rearranged without changing the result because of the commutative property of addition.
Identity property: The property was not explicitly used in this example, but it is worth noting that adding 0 to any number leaves it unchanged (i.e., a + 0 = a).
The properties used to simplify the expression are Associative property of addition, Commutative property of addition, and Distributive property and Identity property of addition. Therefore, the correct option is A, C, E and F.
The properties used to simplify the expression are as follows.
1. Distributive property (E): 4 + 3(9 + 2) = 4 + (3 × 9) + (3 × 2)
This property is applied when a number is multiplied with the sum of two or more numbers. In this case, the number 3 is distributed over the numbers 9 and 2.
2. Identity property of addition (F): 4 + 27 + 6 = 4 + 6 + 27
This property states that adding zero to any number does not change its value. Although this property isn't explicitly shown in the given steps, it is implied by the fact that we can rearrange the terms in the addition without changing their value.
3. Commutative property of addition (C): 4 + 6 + 27 = 10 + 27
This property states that changing the order of numbers in an addition does not change the sum. Here, the numbers 4 and 6 were rearranged to make it easier to add them together.
4. Associative property of addition (A): (10 + 27) = 37
This property states that the grouping of numbers in an addition does not affect the sum. In this case, the parentheses are unnecessary since the numbers were already grouped correctly.
In summary, the properties used to simplify the expression are: A) Associative property of addition, C) Commutative property of addition, and E) Distributive property and F) identity property of addition.
Note: The question is incomplete. The complete question probably is: The real number properties can be used to simplify numerical expressions. in this section, you will identify which properties were used to simplify several expressions. Which properties were used to simplify the following expression? select all that apply.
4 + 3(9 + 2)
4 + (3 × 9) + (3 × 2)
4 + 27 + 6
4 + 6 + 27
10 + 27
37
A) associative property of addition B) associative property of multiplication C) commutative property of addition D) commutative property of multiplication E) distributive property F) identity property of addition G) identity property of multiplication
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PLEASE HELP ME AS SOON AS POSSIBLE WITH EXPLANATIONS PLEASE!!!!!
The statements true of the two-dimensional plane sections that could result from one of these slices made by Misha are B, C, D, E, and F.
What makes a two-dimensional plane sections?A. False. The only two-dimensional plane sections that could result from slicing a cube with a plane are squares or rectangles, but not triangles.
B. True. A plane section that is square could result from one of these slices through the cube.
C. True. A plane section that is rectangular but not square could result from one of these slices through the cube.
D. True. A plane section that is triangular could result from one of these slices through the pyramid.
E. True. A plane section that is square could result from one of these slices through the pyramid.
F. True. A plane section that is rectangular but not square could result from one of these slices through the pyramid.
Therefore, the correct statements are B, C, D, E, and F.
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What are the operations in the equation 4x – 5 = 7? What operations do you need to use to solve for x?
Answer:
x=3, Adding and dividing. (Im not too sure how to answer that question, Are there some options that you learned in class?)
Step-by-step explanation:
4x-5=7
+5 +5
4x=12
/4 /4
x=3
Please upload a picture of a piece of paper with the problem worked out, and draw the graph for extra points, there will be 6 of these, so go to my profile and find the rest, and do the same, for extra points. solve the system using the ELIMINATION method.
The solution to this system of equations are x = 7 and y = -3.
How to solve these system of linear equations?In order to determine the solution to a system of two linear equations, we would have to evaluate and eliminate each of the variables one after the other, especially by selecting a pair of linear equations at each step and then applying the elimination method.
Given the following system of linear equations:
3y = 26 - 5x .........equation 1.
6x + 7y = 21 .........equation 2.
Rewriting in standard form, we have:
5x + 3y = 26
6x + 7y = 21
By multiplying equation 1 by 6 and dividing by 5, we have:
6x + 3.6y = 31.2 .........equation 3.
By subtracting equation 3 from equation 2, we have:
3.4y = -10.2
y = -3.
x = (26 - 3y)/5
x = (26 - 3(-3))/5
x = 7
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Cher is making hotdogs for her coworkers to celebrate their 5 year
anniversary. Hotdogs come in packs of 6, while the buns come in
packs of 10. How many hotdogs should Cher cook to have the
smallest number of hotdogs and hotdog buns?
HELP PLEASE I AM STRUGGLING!!!!!!!!!!!!
Samantha is told that sin (O) = and tan (0)<0, and was asked to determine the value of cos (©).
• She uses the Pythagorean identity and defines it to be sin() + cos2 ( 0 )= 1.
• She substitutes į into the equation for sin (0)
• She then looks at values of sin ( ) and tan () and says that the only quadrant in which sin() is positive and
tan() is negative is the first quadrant.
Samantha determines that the answer is
cos () =
(21)
Samantha determines the answer as cos(θ) = -√(3/4).
Based on the information given, I can help you with this problem. Samantha is correct in using the Pythagorean identity and substitution, but she made an error in identifying the quadrant. Here's the correct process:
1. Given sin(θ) = 1/2 and tan(θ) < 0.
2. The correct quadrant where sin(θ) is positive and tan(θ) is negative is the second quadrant, not the first.
3. Using the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1.
4. Substitute sin(θ) = 1/2 into the equation: (1/2)^2 + cos^2(θ) = 1.
5. Simplify: 1/4 + cos^2(θ) = 1.
6. Solve for cos^2(θ): cos^2(θ) = 3/4.
7. Since we are in the second quadrant, cos(θ) is negative: cos(θ) = -√(3/4).
Your answer: cos(θ) = -√(3/4).
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Find the smallest whole number that is divisible by both 720 and 1575
Answer:
LCM = 2^4 x 3^2 x 5^2 x 7 = 25200
Step-by-step explanation:
Prime factorization of 720:
720 = 2^4 x 3^2 x 5
Prime factorization of 1575:
1575 = 3^2 x 5^2 x 7
Answer:
720 = 2 × 2 × 2 × 2 × 3 × 3 × 5
1,575 = 3 × 3 × 5 × 5 × 7
LCM of 720 and 1,575 =
2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 × 7 = 25,200
The Bayview community pool has a snack stand where Juan works part time he tracks his total sales during each shift last month this box plot shows the results what fraction of Juan’s shifts had a total sales of $225 or more
The fraction of Juan's shifts with a total sales of $225 or more can be found by looking at the box plot.
We can see that the top line of the box represents the third quartile (Q3) which is the value where 75% of the data falls below.
In this case, Q3 is at approximately $250. This means that 75% of Juan's shifts had total sales less than $250. To find the fraction of shifts with sales of $225 or more, we need to determine how many shifts fall within the range of $225 to $250.
Looking at the box plot, we can see that the distance between Q1 and Q3 (the interquartile range) is approximately $100. Therefore, the distance between Q1 and $225 is approximately one-third of the interquartile range or $33.33. So, any shift with total sales of $225 or more would fall within one-third of the distance between Q1 and Q3.
Therefore, the fraction of Juan's shifts with total sales of $225 or more is approximately one-third of 75%, which is 25%.
In summary, approximately 25% of Juan's shifts at the Bayview community pool had total sales of $225 or more, based on the box plot.
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The ratio of three numbers is 6 : 1 : 5. The sum of the numbers is 36. What are the three numbers?
Answer:3,15,18.
Step-by-step explanation:
6:1:5 total ratio =6+1+5=12 so you’ll take all the numbers at different times so 6 will be divided by 12 and multiplied by36 (6/12)36= 18so the first number is nine do the same thing for the next ratio (1/12)36=3 thirdly(5/12)36=15 now add the three numbers to check whether they sum up to36(18+3+15=36)
If the price of a car is $5,900 and has a down payment of 15%, with a tax rate of 8.5%, how much will the amount of the loan need to be for?
Hamid's soccer game will start at 10:00 am but the players must arrive to the field three quarters of an hour early to warm up. the game must end by 1:15
Hamid's soccer game starts at 10:00 am with players warming up 45 minutes earlier and ends by 1:15.
How long is Hamid's soccer game?Hamid's soccer game is scheduled to start at 10:00 am, but the players must arrive at the field 45 minutes early to warm up. This means that they need to be there at 9:15 am. The duration of the game is not given, but we know that it must end by 1:15 pm.
Assuming that the game will last for 90 minutes, it would end at 11:30 am. This would give the players ample time to change, clean up, and leave the field by 12:00 pm. However, if the game were to last longer, say for 2 hours, it would end at 12:00 pm, leaving only 15 minutes for the players to get ready to leave.
Therefore, it is important for the players to play efficiently and within the time allotted so that they have enough time to change and leave before the deadline. It is also important for the players to arrive at the field on time to ensure that they have enough time to warm up and prepare for the game.
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What is the average rate of change of the function g(x)=6x from x=-1 to x=3? show your work or explain how you obtained your response
The average rate of change of the function g(x)=6x from x=-1 to x=3 is found to be 6.
The function g(x) = 6x describes a relationship between x and the value of 6 times x. We want to find the average rate of change of this function from x = -1 to x = 3. The average rate of change tells us the average amount by which the function changes per unit of change in x over this interval.
In this case, by using the function g(x) = 6x and evaluating it for x = 3 and x = -1, a difference of 18 - (-6) = 24 is found. The difference in x's values is equal to 3 - (-1) = 4. We divide these to get an average rate of change of 6.
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(1 point) Write an equivalent integral with the order of integration reversed g(y) I hope F(x,y) dydt = F(x,y) dedy f(y) a = b= f(y) = g(y) =
The missing values are:
a = 0b = 1c = 1f(y) = yg(y) = 2 - yh(y) = 0k(y) = yGiven Integral:
[tex]\int\limits^1_0 \int\limits^{2-x}_x {F(x,y)} \, dydx = \int\limits^b_a \int\limits^{g(y)}_{f(y)} {F(x,y)} \ dxdy + \int\limits^c_b \int\limits^{h(y)}_{k(y)} {F(x,y)} \ dxdy \\[/tex]
To write the equivalent integral with the order of integration reversed, express the limits of integration and functions appropriately.
Reversed integral:
[tex]\int\limits^b_a \int\limits^{g(y)}_{f(y)} {F(x,y)} \ dxdy + \int\limits^c_b \int\limits^{h(y)}_{k(y)} {F(x,y)} \ dxdy \\[/tex]
Now, let's determine the values of the variables:
a = 0: The lower limit of the outer integral remains the same as the original integral.
b = 1: The upper limit of the outer integral also remains the same as the original integral.
c = 1: The upper limit of the second inner integral is determined by the limits of integration of the original integral, which is 1.
f(y) = y: The lower limit of the first inner integral is the same as the original integral, which is y = x.
g(y) = 2 - y: The upper limit of the first inner integral is determined by the limits of integration of the original integral, which is 2 - x.
h(y) = 0: The lower limit of the second inner integral remains the same as the original integral.
k(y) = y: The upper limit of the second inner integral remains the same as the original integral.
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Express the area of the entire rectangle. your answer should be a polynomial in standard form. x+6 x+2 area=
The formula for the area of a rectangle is length times width, or in this case (x+6)(x+2) or x^2 + 8x + 12 in standard form.
To simplify this expression, we can use the distributive property to multiply the two binomials:
(x+6)(x+2) = x(x+2) + 6(x+2)
= x² + 2x + 6x + 12
= x² + 8x + 12
To express the area of the entire rectangle, we need to multiply the length (x + 6) by the width (x + 2). This will give us a polynomial in standard form.
Step 1: Write down the expression for the area of the rectangle.
Area = (x + 6)(x + 2)
Step 2: Use the distributive property (also known as the FOIL method) to expand the expression.
Area = x(x + 2) + 6(x + 2)
Step 3: Continue to expand and simplify the expression.
Area = (x² + 2x) + (6x + 12)
Step 4: Combine like terms.
Area = x² + 8x + 12
So the area of the entire rectangle is expressed as the polynomial x² + 8x + 12 in standard form.
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HELP ME PLEASE I WILL GIVR BRAINLIEST TO THE FASTED CORRECT ANSWER PLEASE HELP ME FAST AND TY
What is the slope of y = 3x - 2?
Miranda ran 4 miles in 28 minutes how many miles does miranda run in 2 minutes
Answer:
We know that Miranda ran 4 miles in 28 minutes. Let's set up the proportion:
4 miles / 28 minutes = x miles / 2 minutes
To solve for x, we can cross-multiply:
28 minutes * x miles = 4 miles * 2 minutes
28x = 8
Now, let's solve for x by dividing both sides of the equation by 28:
x = 8 / 28
x = 0.2857
Therefore, Miranda runs approximately 0.2857 miles in 2 minutes.
In ARST, r = 58 cm, m/S=48° and m/T=29°. Find the length of s, to the nearest
centimeter.
The value of length 's' is 44.3 cm
What is sine rule?The sine rule states that if a, b and c are the lengths of the sides of a triangle, and A, B and C are the angles in the triangle; with A opposite a, etc., then a/sinA=b/sinB=c/sinC.
The measure of angle R = 180-( 48+29)
R = 180- 77
R = 103°
sinR/ r = sinS/s
sin103 / 58 = sin48/s
s × sin103 = 58 × sin48
s × 0.974 = 43.1
s = 43.1/0.974
s = 44.3 cm
therefore the value of length is is 44.3 cm
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A Film crew is filming an action movie where a helicopter needs to pick up a stunt actor located on the side of a canyon actor is 20 feet below the ledge of the canyon the helicopter is 30 feet above the canyon. Which of the following expressions represents the length of rope that needs to be lowered from the helicopter to reach the stunt actor
The expression that represents the length of rope that needs to be lowered is 30 - -20
Which expression represents the length of rope that needs to be loweredFrom the question, we have the following parameters that can be used in our computation:
canyon actor is 20 feet below the ledge of the canyon Helicopter is 30 feet above the canyonUsing the above as a guide, we have the following:
Length of rope = helicopter - canyon
So, we have
Length of rope = 30 - -20
Evaluate
Length of rope = 50
Hence, the length of rope is 50 feet
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Match each equation to its graph and table representation
Answer:
The first one is B & H
2nd one is A & G
3rd one is D & E
4th one is C & F
Step-by-step explanation:
A. johnny translated abcd 3 units to the right and 4 units up to a new position, efgh. draw and label efgh.
b. tom rotated abcd to a new position, ijkl, 90º clockwise about the origin, o. draw and label ijkl.
c. tony placed a smaller car, represented as mnop, on the coordinate plane. mnop is a dilation of abcd with its center at the origin and a scale factor of -0.5. draw and label mnop.
A. To obtain the position of EFGH, Johnny translated ABCD by 3 units to the right and 4 units up. To draw and label EFGH, simply shift each vertex of ABCD by this translation vector (3, 4).
B. Tom rotated ABCD by 90º clockwise about the origin, O, to get the position of IJKL. To draw and label IJKL, rotate each vertex of ABCD 90º clockwise around the origin. This can be achieved by switching the x and y coordinates of each vertex and negating the new x value.
C. Tony placed a smaller car, MNOP, on the coordinate plane. MNOP is a dilation of ABCD with its center at the origin and a scale factor of -0.5. To draw and label MNOP, multiply the coordinates of each vertex of ABCD by the scale factor -0.5, keeping the origin as the center.
How do you do this problem?
Answer: 135 and 45
Step-by-step explanation:
We can read off from these equations the gradients of the two lines: (3) and (-2).
Then we quote the trigonometric identity tan(A-B) = [tan(A)-tan(B)] / [1+tan(A)tan(B)]
Substituting tan(A)=3 and tan(B)=-2 gives tan(A-B) = [(3)-(-2)] / [1+(3)(-2)] = 5/-5 = -1
So A-B = 135°.
That is the obtuse angle between the two lines, so the acute angle is 45°.