Using more advanced technology, a team of workers began to produce 6 more parts per hour than before. In six hours, the team produced 120% of what they had previously been able to produce in eight hours. How many parts per hour was the team producing prior to switching to the new technology?
Answer: Therefore, the team was producing 10 parts per hour prior to switching to the new technology.
Step-by-step explanation:Let's denote the number of parts produced per hour before the technology upgrade by x.
After the upgrade, the team produces 6 more parts per hour than before, so their new production rate is x + 6 parts per hour.
In 8 hours, the team produces 8x parts in total.
In 6 hours with the new technology, the team produces 120% of what they previously produced in 8 hours, or 1.2(8x) = 9.6x parts in total.
We can set up an equation based on the information above:
6(x + 6) = 9.6x
Simplifying the equation:
6x + 36 = 9.6x
Subtracting 6x from both sides:
36 = 3.6x
Dividing both sides by 3.6:
x = 10
Evaluate the definite integrals ∫(9x^2 - 4x - 1)dx =
Definite integral of ∫(9x^2 - 4x - 1)dx from a to b is 3(b^3 - a^3) - 2(b^2 - a^2) - (b - a).
To evaluate the definite integral ∫(9x^2 - 4x - 1)dx, you need to first find the indefinite integral (also known as the antiderivative) of the function 9x^2 - 4x - 1. The antiderivative is found by applying the power rule of integration to each term separately:
∫(9x^2)dx = 9∫(x^2)dx = 9(x^3)/3 = 3x^3
∫(-4x)dx = -4∫(x)dx = -4(x^2)/2 = -2x^2
∫(-1)dx = -∫(1)dx = -x
Now, sum these results to obtain the antiderivative:
F(x) = 3x^3 - 2x^2 - x
∫(9x^2 - 4x - 1)dx from a to b = F(b) - F(a)
To evaluate the definite integral ∫(9x^2 - 4x - 1)dx =, we need to use the formula for integrating polynomials. Specifically, we use the power rule of integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration.
Using this formula, we integrate each term in the given expression separately. Thus, we have:
∫(9x^2 - 4x - 1)dx = (9∫x^2 dx) - (4∫x dx) - ∫1 dx
= 9(x^3/3) - 4(x^2/2) - x + C
= 3x^3 - 2x^2 - x + C
Next, we need to evaluate this definite integral. A definite integral is an integral with limits of integration, which means we need to substitute the limits into the expression we just found and subtract the result at the lower limit from the result at the upper limit. Let's say our limits are a and b, with a being the lower limit and b being the upper limit. Then, we have:
∫(9x^2 - 4x - 1)dx from a to b = [3b^3 - 2b^2 - b] - [3a^3 - 2a^2 - a]
= 3(b^3 - a^3) - 2(b^2 - a^2) - (b - a)
Therefore, the definite integral of ∫(9x^2 - 4x - 1)dx from a to b is 3(b^3 - a^3) - 2(b^2 - a^2) - (b - a).
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The graph shows the height of a scratch on the edge of a circular gear.
Which function is the best model for the height of the scratch?
a. h(t) = 3.5 sin (π t) + 1.5
b. g(t) = 1.5 sin (π t) +3.5
c. h(t) = 1.5 sin (2 π t) + 3.5
d. h(t) = 1.5 sin (π/2 t) + 3.5
Answer:
b. g(t) = 1.5 sin (π t) +3.5
Step-by-step explanation:
You want to choose the function that has the given graph.
Test pointsAt t = 0, the graph shows a value of 3.5. The sine of 0 is 0, so this eliminates choice A.
At t = 1/2, the graph shows a value of 5. The values given by the different formulas are ...
b. g(1/2) = 1.5·sin(π/2) +3.5 = 5 . . . . . matches the graph
c. h(1/2) = 1.5·sin(π) + 3.5 = 3.5 . . . . no match
d. h(1/2) = 1.5·sin(π/4) +3.5 = 0.75√2 +3.5 . . . . no match
__
Additional comment
The horizontal distance for one period of the graph (from peak to peak, for example) is T = 2 seconds. If the sine function is sin(ωt), then the value of ω is ...
ω = 2π/T = 2π/2 = π
This tells you the function g(t) = 1.5·sin(πt)+3.5 is the correct choice.
(a) Find a counterexample which shows that WAT is not true if we replace the closed interval [a,b] with the open interval (a,b).(b) What happens if we replace [a,b] with the closed set [a,\infty). Does the theorem hold?
(a) WAT is not true for the open interval (0,1) with function f(x) = 1/x.
(b) WAT holds for the closed set [a,∞) with any continuous function f(x).
(a) The Weierstrass Approximation Theorem (WAT) is not true if we replace the closed interval [a,b] with the open interval (a,b). A counterexample is the function f(x) = 1/x on the open interval (0,1). This function is continuous on (0,1) but it is not uniformly continuous, so it cannot be uniformly approximated by a polynomial.
(b) The Weierstrass Approximation Theorem holds if we replace [a,b] with the closed set [a,∞). That is, if f(x) is a continuous function on [a,∞), then for any ε > 0, there exists a polynomial p(x) such that |f(x) - p(x)| < ε for all x in [a,∞). The proof is similar to the proof of the original theorem using the Bernstein polynomials.
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Abc company’s budgeted sales for june, july, and august are 12,800, 16,800, and 14,800 units, respectively. abc requires 30% of the next month’s budgeted unit sales as finished goods inventory each month. budgeted ending finished goods inventory for may is 3,840 units. each unit that abc company produces uses 2 pounds of raw material. abc requires 25% of the next month’s budgeted production as raw material inventory each month.
The budgeted ending raw material inventory for May is 2,560 pounds, calculated by taking 25% of the next month's budgeted production (12,800 units) multiplied by 2 pounds per unit.
To solve this problem, we need to calculate the budgeted production and raw material inventory for June, July, and August.
For June:Budgeted production = 12,800 units + 30% * 16,800 units = 17,440 units
Raw material inventory = 25% * 17,440 units * 2 pounds = 8,720 pounds
For July:Budgeted production = 16,800 units + 30% * 14,800 units = 20,840 units
Raw material inventory = 25% * 20,840 units * 2 pounds = 10,420 pounds
For August:Budgeted production = 14,800 units + 30% * 20,840 units = 20,632 units
Raw material inventory = 25% * 20,632 units * 2 pounds = 10,316 pounds
To find the budgeted ending finished goods inventory for June, we need to subtract the budgeted sales for June from the budgeted production for June and add the budgeted ending finished goods inventory for May:
Budgeted ending finished goods inventory for June = 17,440 units - 12,800 units + 3,840 units = 8,480 units
Similarly, we can find the budgeted ending finished goods inventory for July and August:
Budgeted ending finished goods inventory for July = 20,840 units - 16,800 units + 8,480 units = 12,520 units
Budgeted ending finished goods inventory for August = 20,632 units - 14,800 units + 12,520 units = 18,352 units
Therefore, the budgeted ending finished goods inventory for June, July, and August are 8,480 units, 12,520 units, and 18,352 units, respectively. The budgeted raw material inventory for June, July, and August are 8,720 pounds, 10,420 pounds, and 10,316 pounds, respectively.
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Please help it due soon and the answer is meant to be in kg
Answer: 20 kg
Step-by-step explanation:
You follow the line of best fit until 50cm
Then you trace across and look at the x-axis.
There you will find that the dog will be 20kg at 50cm using the line of best fit.
In a city of 72,500 people, a simple random sample of four households is selected from the 25,000 households in the population to estimate the average cost on food per household for a week. the first household in the sample had 4 people and spent a total of $150 in food that week. the second household had 2 people and spent $100. the third, with 4 people, spent $200. the fourth, with 3 people, spent $140.
required:
identify the sampling units, the variable of interest, and any auxiliary info mation associated with the units.
In this scenario, the sampling units are four households, the variable of interest is the average food cost, and auxiliary information associated with the units is the number of people in each household and total food cost.
Sampling Units: The sampling units are the four households selected from the 25,000 households in the population.
They are as follows:
1. Household with 4 people that spent $150 on food
2. Household with 2 people that spent $100 on food
3. Household with 4 people that spent $200 on food
4. Household with 3 people that spent $140 on food
Variable of Interest: The variable of interest is the average cost on food per household for a week.
Auxiliary Information: The auxiliary information associated with the units includes the number of people in each household and the total amount spent on food for that week.
To estimate the average cost on food per household for a week, follow these steps:
1. Calculate the total cost on food for all four households: $150 + $100 + $200 + $140 = $590
2. Divide the total cost by the number of households: $590 / 4 = $147.50
So, the estimated average cost on food per household for a week is $147.50.
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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°.
What three-dimensional figure is formed when the triangle shown is rotated around the dashed line?
A. cone
B. cylinder
C. double cone
D. hemisphere
Answer: C
Step-by-step explanation: after rotating, if you split it in half horizontally, you have two cones
The three-dimensional figure formed when the triangle is rotated around the dashed line through B and C is a cone.
What is a cone?A cone is a three-dimensional geometric form with a flat base and a smooth, tapering apex or vertex. A cone is made up of a collection of line segments, half-lines, or lines that link the base's points to the apex, which is a common point on a plane that does not include the base.
When we rotate a two-dimensional shape around an axis, we create a three-dimensional solid. This process is known as "revolution" or "rotational symmetry".
In this particular case, we have a triangle that can be rotated around the line segment that connects points B and C. If we were to rotate the triangle around this axis, we would create a three-dimensional solid. To figure out what kind of solid this is, we can think about the cross-sections that would be created if we were to slice through the solid perpendicular to the axis of rotation.
If we were to slice through the solid perpendicular to the axis of rotation, we would get a circle. This means that the solid created by rotating the triangle is a cylinder.
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At a print shop reams of printer paper are stored in boxes in a closet. Each box contains 12 reams of printer paper. A worker uses 4 reams from 1 of the boxes. Which function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet?
The function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet is y = 12x - 4
Let's start by considering the initial amount of printer paper in the closet before any boxes are used. Since each box contains 12 reams of printer paper, if there are x boxes in the closet, then the total number of reams of paper is given by 12x.
Now, if a worker uses 4 reams from one of the boxes, then the total number of reams of paper remaining in the closet is (12x - 4). If we define y as the total number of reams of paper remaining in the closet, then we have:
y = 12x - 4
This function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet.
As x increases, the total number of reams of paper in the closet increases as well. However, each time a worker uses 4 reams of paper from a box, the total number of reams of paper in the closet decreases by 4.
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HELP ME PLEASE I DON'T UNDERSTAND
Answer:
34/73
Step-by-step explanation:
37 + 34 + 2 = number of customers = 73
73 is our denominator.
34 is the number of people who used a credit card.
34 is our numerator.
Put the two together, and you get 73! Enjoy!
If the points a,b and c have the coordinates a(5,2) , b(2,-3) and c(-8,3) show that the triangle abc is a right angled triangle
Points a,b and c satisfied the Pythagoras theorem. Thus, the triangle abc is a right angled triangle.
Define about the right angled triangle:Every triangle has inner angles that add up to 180 degrees. A right angle and a right triangle are both formed when one of their internal angles is 90 degrees.
The internal 90° angle of right triangles is denoted by a little square in the vertex. The complimentary angles of the other two sides of a right triangle sum up to 90 degrees.The triangle's legs, which are typically denoted by the letters a and b, are the sides that face the complimentary angles.Given coordinates :
a(5,2) , b(2,-3) and c(-8,3).
Find the distance between the points using the distance formula:
d = √[(x2 - x1)² + (y2 - y1)²]
ab = √[(2 - 5)² + (- 3 - 2)²]
ab = √[(-3)² + (- 5)²]
ab = √[9 + 25]
ab = √34
ab² = 34
bc = √[(2 + 8)² + (- 3 - 3)²]
bc = √[(10)² + (- 6)²]
bc = √[100 + 36]
bc = √136
bc² = 136
ac = √[(-8 - 5)² + (3 - 2)²]
ac = √[(-13)² + (1)²]
ac = √[169 + 1]
ac = √170
ac² = 170
Now,
(ac)² = (bc)² + (ab)²
170 = 136 + 24
170 = 170
This, points a,b and c satisfied the Pythagoras theorem. Thus, the triangle abc is a right angled triangle.
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A group of friends wants to go to the amusement park. They have $100. 25 to spend
on parking and admission. Parking is $17. 75, and tickets cost $13. 75 per person,
including tax. Which equation could be used to determine p, the number of people
who can go to the amusement park?
100. 25 = 13. 75p + 17. 75
Op=
100. 25-13. 75
17. 75
Submit Answer
13. 75(p+17. 75) = 100. 25
O p =
17. 75-100. 25
13. 75
The correct equation to determine the number of people (p) who can go to the amusement park is: 100.25 = 13.75p + 17.75.
Here's the step-by-step explanation:
1. The total amount they have to spend is $100.25.
2. The cost of parking is $17.75, which is a one-time expense.
3. The cost of admission per person is $13.75.
To find out how many people can go, you need to account for both the parking cost and the cost of tickets for each person. Therefore, the equation is:
100.25 (total amount) = 13.75p (cost per person times the number of people) + 17.75 (cost of parking)
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The measures of the angles of a triangle are shown in the figure below. Solve for x.
(2x+16) 48degrees
Answer:
x = 16
Step-by-step explanation:
(2x + 16) = 48
Subtract 16 with the positive 16 to cancel the numbers.
Subtract 16 with 48.
2x = 32
divide 32 by 2 to isolate the x.
32/2 = 16
x = 16
Use Newton's method to approximate a root of the equation5sin(x)=xas follows. Letx1=1 be the initial approximation. The second approximationx2 is and the third approximationx3 is
The second approximation x2 is approximately 1.112141637097, and the third approximation x3 is approximately 1.130884826739.
Newton's method to approximate a root of the equation 5sin(x) = x.
We are given the initial approximation x1 = 1. To find the second approximation x2 and the third approximation x3, we need to follow these steps:
Step 1: Write down the given function and its derivative. f(x) = 5sin(x) - x f'(x) = 5cos(x) - 1
Step 2: Apply Newton's method formula to find the next approximation. x_{n+1} = x_n - f(x_n) / f'(x_n)
Step 3: Calculate the second approximation x2 using x1 = 1. x2 = x1 - f(x1) / f'(x1) x2 = 1 - (5sin(1) - 1) / (5cos(1) - 1) x2 ≈ 1.112141637097
Step 4: Calculate the third approximation x3 using x2. x3 = x2 - f(x2) / f'(x2) x3 ≈ 1.112141637097 - (5sin(1.112141637097) - 1.112141637097) / (5cos(1.112141637097) - 1) x3 ≈ 1.130884826739
So, the second approximation x2 is approximately 1.112141637097, and the third approximation x3 is approximately 1.130884826739.
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• use the regression function from the previous step as a mathematical model for the demand function
(e.g. d(p)) and find the general expression for the elasticity of demand:
ep)
To find the general expression for the elasticity of demand (e_p), we need to differentiate the demand function with respect to price (p) and multiply it by the ratio of price to quantity (p/q). The elasticity of demand measures the responsiveness of quantity demanded to changes in price.
The general expression for elasticity of demand (e_p) can be calculated as:
e_p = (dQ/dp) * (p/Q)
Where dQ/dp represents the derivative of the demand function with respect to price, and Q represents the quantity demanded.
The elasticity of demand helps us understand how sensitive the quantity demanded is to changes in price. If e_p is greater than 1, demand is considered elastic, meaning that quantity demanded is highly responsive to price changes. If e_p is less than 1, demand is inelastic, indicating that quantity demanded is less responsive to price changes.
In conclusion, the general expression for the elasticity of demand (e_p) is calculated by taking the derivative of the demand function with respect to price and multiplying it by the ratio of price to quantity. This measure helps determine the responsiveness of quantity demanded to changes in price.
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Find the sum of the first 10 terms of the following series, to the nearest integer.
8,20/3,50/9
The sum of the first 10 terms of the given series 8,20/3,50/9... is 140.
Given series: 8,20/3,50/9...
The given series is not in a standard form, but it appears to be an arithmetic sequence with a common difference of 4/3. To check this, we can find the difference between consecutive terms:
20/3-8=4/3
50/9-20/3=4/3
Thus, the common difference is indeed [tex]\frac{4}{3}[/tex].
We notice that each term of the series can be written as:
a_n=a+(n-1)d
a_n=8+(n-1)(4/3)
where n is the index of the term, and 4/3 is the common difference between the consecutive terms.
To find the sum of the first 10 terms of the series, we use the formula for the sum of an arithmetic series:
S=(n/2)[2a_1+(n-1)d]
where S is the sum of the series, a_1 is the first term of the series, d is the common difference, and n is the number of terms to be added.
Substituting the given values, we get:
S=(10/2)[2*8+(10-1)(4/3)]
Simplifying the expression:
S=5[16+9(4/3)]
S=5[16+12]=5(28)=140
Therefore, the sum of the first 10 terms of the series is 140.
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The profit in dollars from the sale of x expensive watches is P(x) = 0.08x² - 5x + 6x0.2 - 5200 Find the marginal profit when (a) x = 300. (b) x = 2000, (c) X = 5000, and (d) x = 12,000.
The marginal profit in dollars for the sale of expensive watches when approximately $1912.61.
Find the marginal profit in dollars from the sale?
We need to find the marginal profit in dollars from the sale of x expensive watches for the given profit function P(x) = 0.08x² - 5x + 6x^0.2 - 5200 when (a) x = 300, (b) x = 2000, (c) x = 5000, and (d) x = 12,000.
Find the derivative of the profit function P(x), which represents the marginal profit.
P'(x) = dP(x)/dx = 0.16x - 5 + (6 * 0.2 * x^(-0.8))
Calculate the marginal profit for each specified value of x:
x = 300:
P'(300) = 0.16(300) - 5 + (6 * 0.2 * 300^(-0.8)) ≈ 42.57
x = 2000:
P'(2000) = 0.16(2000) - 5 + (6 * 0.2 * 2000^(-0.8)) ≈ 317.52
x = 5000:
P'(5000) = 0.16(5000) - 5 + (6 * 0.2 * 5000^(-0.8)) ≈ 794.57
x = 12,000:
P'(12,000) = 0.16(12,000) - 5 + (6 * 0.2 * 12,000^(-0.8)) ≈ 1912.61
So, the marginal profit in dollars for the sale of expensive watches when (a) x = 300 is approximately $42.57, (b) x = 2000 is approximately $317.52, (c) x = 5000 is approximately $794.57, and (d) x = 12,000 is approximately $1912.61.
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Geometry
three squares with areas of 64, 225, and 289 square units are arranged so that when their vertices coincide a triangle is formed. find the area of that triangle.
please explain how you solved this along with the answer.
Answer:
The area of the largest square is 289 square units, because it is the sum of areas of the two smaller squares, 64 square units and 225 square units.
Step-by-step explanation: JUST PASSED IT ON STUDY ISLAND 100% CORRECT ANSWER
HELPPPPPPP PLEASEEEE
Answer:
The first box and whisker plot
Step-by-step explanation:
A box and whisker plot gives you the five number summary for a set of data. The five number summary is
The minimum/lowest value (looks like the top of capital T turned sideways and is the leftmost part of the box-and-whisker plot The first quartile or Q1, representing 25% of the data (the first point represented in the "box" of the plot and serves as an endpoint of the box)The median or Q2, representing 50%/the middle of the data (the line that splits the box into two parts/the line in the middle of the box)The third quartile or Q3, representing 75% of the data (the last point represented in the "box" of the plot and serves as another endpoint of the box)The maximum/highest value (also looks like the top of capital T turned sideways and is the rightmost part of the box-and-whisker plotMaximum and minimum:
We know from the data that the minimum value is 100 and the maximum value is 200. However, because both boxes available as answer choices have the correct minimum and maximum, we'll need to find more data.
Median:
We can start finding the median first by arranging the data from the least to greatest. Then, we find the middle of the data. Because there are 9 points and 9 is odd, we know that there will be 4 points to the left of the median and 4 points to the right of the median:
100, 100, 120, 120, 150, 165, 180, 180, 200
150 has 4 numbers both on its left and right sides so its the median.
Because both of the plots available as answer choices have the correct median, we we'll need to find more data.
First Quartile/Q1:
In order to find Q1, we must find the middle number of the four numbers to the left of the median.
Because we have an even number of points, we will get two middle numbers, 100 and 120. To find the middle of all four points, we average these two numbers:
(100 + 120) / 2 = 220 / 2 = 110
Only the first box has the accurate Q1 value, so it's our answer.
We don't have to find Q3, since both boxes have the correct Q3, but only the first box has the correct minimum, correct Q1, correct median, correct Q3, correct maximum.
Find the surface area of the composite figure.
2 in.
4 in.
9 in.
SA
=
7 in.
2
[?] in.²
4 in.
4 in.
If you'd like,
you can use a
calculator.
Answer:
236 in²
Step-by-step explanation:
You want the surface area of the figure comprised of two cuboids.
AreaThe surface area of the figure will be the sum of the total surface area of the purple cuboid, plus the lateral surface area of the yellow cuboid.
SA = 2(LW +H(L +W)) + Ph
SA = 2(9·4 +2(9 +4)) +(4·4)(7) = 236 . . . . square inches
The surface area of the composite figure is 236 square inches.
__
Additional comment
You can consider the face on the right side to be equal in area to the area of the purple cuboid that is covered by the yellow one. So, figuring the total area of the purple cuboid effectively includes the area of the face on the right side.
Then the remaining part of the area of the yellow cuboid is the area of the four 7×4 rectangles that are its lateral area.
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Find the new coordinates for the image under the given dilation. Rhombus WXYZ with vertices W(1, 0), X (4,-1), Y(5,-4), and Z(2, -3): k = 3. W' (.) x' (,) X' Y'(,) Z' (
the new coordinates of the rhombus W'X'Y'Z' after a dilation with scale factor k=3 are: [tex]W'(3,0), X'(12,-3), Y'(15,-12), Z'(6,-9)[/tex]
What are the coordinates?To find the new coordinates of the image after dilation, we need to multiply the coordinates of each vertex by the scale factor k = 3.
Let's start with vertex W(1,0):
Multiply the x-coordinate by [tex]3: 1 *\times 3 = 3[/tex]
Multiply the y-coordinate by [tex]3: 0 \times 3 = 0[/tex]
So the new coordinates of W' are [tex](3,0).[/tex]
Next, let's look at vertex X(4,-1):
Multiply the x-coordinate by [tex]3: 4 \times 3 = 12[/tex]
Multiply the y-coordinate by [tex]3: -1 \times 3 = -3[/tex]
So the new coordinates of X' are [tex](12,-3).[/tex]
Now for vertex Y(5,-4):
Multiply the x-coordinate by [tex]3: 5 \times 3 = 15[/tex]
Multiply the y-coordinate by [tex]3: -4 \times3 = -12[/tex]
So the new coordinates of Y' are [tex](15,-12).[/tex]
Finally, let's consider vertex Z(2,-3):
Multiply the x-coordinate by [tex]3: 2 \times 3 = 6[/tex]
Multiply the y-coordinate by [tex]3: -3 \times3 = -9[/tex]
So the new coordinates of Z' are [tex](6,-9)[/tex] .
Therefore, the new coordinates of the rhombus [tex]W'X'Y'Z'[/tex] after a dilation with scale factor k=3 are:
[tex]W'(3,0)[/tex]
[tex]X'(12,-3)[/tex]
[tex]Y'(15,-12)[/tex]
[tex]Z'(6,-9)[/tex]
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Let f(2)= 1 / x² + root x, is it converge or diverge?
To determine whether the function f(2) converges or diverges, we need to evaluate the limit of the function as x approaches 2. We can rewrite the function as:
f(2) = 1 / (x² + √x) = 1 / (x² + x^(1/2))
As x approaches 2, both x² and x^(1/2) approach 2, so we can substitute 2 for both of these terms:
f(2) = 1 / (2² + 2^(1/2)) = 1 / (4 + 1.414) ≈ 0.176
Therefore, f(2) converges to a finite value of approximately 0.176, and does not diverge.
Based on the given information, let's analyze the function f(x) = 1 / (x² + √x). To determine if the function converges or diverges, we can examine its behavior as x approaches infinity.
As x gets larger, both x² and √x increase, but x² increases at a much faster rate. Therefore, the denominator (x² + √x) will become larger and larger as x approaches infinity. Consequently, the value of the function f(x) = 1 / (x² + √x) will approach 0.
Since the function approaches 0 as x goes to infinity, we can conclude that the function f(x) = 1 / (x² + √x) converges.
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About 20 years ago, a mathematician noted that his dog, when retrieving a
frisbee in a lake, would run parallel to the shore for quite some distance, and then jump into the water and
swim straight for the frisbee. She would not enter the lake immediately, nor would she wait until she was on
the point on the shore closest to the frisbee. Pennings theorized that the dog entered the water at the point
that would minimize the total length of time it takes to reach the frisbee. Suppose that the dog runs at 13
mph along the shore of the lake but swims at only 4. 3 mph in the water. Further, suppose that the frisbee is
in the water 60 feet off shore and 220 feet down the shoreline from the dog. Suppose that the dog enters the
water after running x feet down the shoreline and then enters the water. Compute the total length of time, T,
it will take for the dog to reach the frisbee. Next, determine a natural closed interval that limits reasonable
values of x. Finally, find the value of x that will minimize the time, T, that it takes for the dog to retrieve the
frisbee
a. The total length of time, T, it will take for the dog to reach the frisbee is 143.22
b. A natural closed interval that limits reasonable values of x is [0, 220] is a reasonable closed interval for x.
c. The value of x that will minimize the time, T, that it takes for the dog to retrieve the frisbee is 143.22
Let's start by breaking down the problem into two parts: the time it takes for the dog to run along the shore, and the time it takes for the dog to swim in the water. Let's call the distance the dog runs along the shore "d1" and the distance the dog swims in the water "d2".
To find d1, we can use the Pythagorean theorem:
d1 = sqrt(x^2 + 60^2)
To find d2, we can use the fact that the total distance the dog travels is equal to 220 feet:
d2 = 220 - x
Now we can use the formulas for distance, rate, and time to find the total time it takes for the dog to retrieve the frisbee:
T = d1/13 + d2/4.3
Substituting our expressions for d1 and d2, we get:
T = [sqrt(x^2 + 3600)]/13 + (220 - x)/4.3
To find the value of x that minimizes T, we can take the derivative of T with respect to x, set it equal to zero, and solve for x:
dT/dx = x/13sqrt(x^2 + 3600) - 1/4.3 = 0
Multiplying both sides by 13sqrt(x^2 + 3600), we get:
x = (13/4.3)sqrt(x^2 + 3600)
Squaring both sides and solving for x, we get:
x ≈ 143.22
So the dog should enter the water after running about 143.22 feet down the shoreline to minimize the total time it takes to retrieve the frisbee.
To check that this is a minimum, we can take the second derivative of T with respect to x:
d^2T/dx^2 = (13x^2 - 46800)/(169(x^2 + 3600)^(3/2))
Since x^2 and 3600 are both positive, the numerator is positive when x is not equal to zero, and the denominator is always positive. Therefore, d^2T/dx^2 is always positive, which means that x = 143.22 is indeed the value that minimizes T.
As for the natural closed interval that limits reasonable values of x, we know that x has to be greater than zero (since the dog needs to run at least some distance along the shoreline before entering the water), and it has to be less than or equal to 220 (since the frisbee is 220 feet down the shoreline from the dog). So the interval [0, 220] is a reasonable closed interval for x.
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GEOMETRY HELP COSINE, SINE TANGENT
please help y’all i have no idea what i am doing
Answer:
31.33 degrees
Step-by-step explanation:
The question is asking to find angle x. You can use sine to find out x because sine is opposite/hypotenuse but since you are finding an angle measurement, it would be to the power of -1. So:
sine^-1=13/25
31.33
Robert, by 3/4 pound of Grace, and divided into six equal portions. What is the way of each portion
Each portion of Grace weighs 1/8 pound.
What is weight?It gauges how much gravity is pulling on a body.
If Robert has 3/4 pound of Grace and he wants to divide it into six equal portions, we can find the weight of each portion by dividing 3/4 by 6:
(3/4) / 6 = (3/4) * (1/6) = 1/8
So each portion of Grace weighs 1/8 pound.
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In parallelogram best, diagonals bs and et bisect each other at o.
1. if es = 10cm, how long is bt?
2. if be = 13cm, how long is ts?
3. if eo = 6cm and so = 7cm, what is the length of et? bs?
4. if et + bs = 18cm and so = 5cm, find et and bs.
When the parallelogram, diagonals bs and et bisect at each other at o, we get the following answers:
1. In a parallelogram, the diagonals bisect each other. So, if ES = 10 cm, then EO = OS = 5 cm. Since EO and OS are half of the diagonal ET, then ET = EO + OS = 5 cm + 5 cm = 10 cm. Similarly, diagonal BT will also be equal to 10 cm, as it has the same length as diagonal ET.
2. In a parallelogram, opposite sides are equal. So, if BE = 13 cm, then TS = 13 cm, as they are opposite sides.
3. If EO = 6 cm and SO = 7 cm, then the length of diagonal ET is EO + OS = 6 cm + 7 cm = 13 cm. Since the diagonals of a parallelogram are equal, the length of diagonal BS will also be 13 cm.
4. If ET + BS = 18 cm and SO = 5 cm, we can use the fact that diagonals bisect each other to find ET and BS. Let EO = x cm. Then, ET = 2x cm and BS = 2(5-x) cm. Now, we can set up the equation: 2x + 2(5-x) = 18. Solving for x, we get x = 4 cm. So, ET = 2x = 8 cm and BS = 2(5-x) = 10 cm.
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Segments OT and OV are?
True, The more variables that are deployed to segment a market, the more useful is the resulting segmentation likely to be.
We have,
The factors which are be used to member a request are the segmentation variables. Common variables include demographic, geographic, psychographics and behavioral considerations.
Quantifiable population characteristics, similar as age, gender, income, education, family situation.
The primary ideal of segmentation is to identify guests with analogous attributes, and to find which parts of guests that are seductive from a profit perspective.
Understanding the request segmentation allows marketers to produce a more effective and effective marketing blend.
Hence, True, The more variables that are deployed to segment a market, the more useful is the resulting segmentation likely to be.
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complete question:
The more variables that are deployed to segment a market, the more useful is the resulting segmentation likely ot be.[See p.104]Group of answer choicesTrueFalse
What is the area of the curved surface of a right circular cone of radius 15 and height 8? The area of the curved surface is | | units. (Type an exact answer in terms of π.)
Curved surface area of cone: 255π or approx. 801.41 sq units with radius 15 and height 8.
The curved surface area of a right circular cone can be calculated using the formula:
A = πrℓ
where A is the area of the curved surface,
r is the radius of the base of the cone, and
ℓ is the slant height of the cone.
To find the slant height, we can use the Pythagorean theorem:
ℓ² = r² + h²
where h is the height of the cone.
Substituting the given values, we get:
ℓ² = 15² + 8²
ℓ² = 225 + 64
ℓ² = 289
ℓ = √289
ℓ = 17
Now, substituting the values of r and ℓ in the formula for curved surface area, we get:
A = πrℓ
A = π(15)(17)
A = 255π
Therefore, the area of the curved surface of the cone is 255π square units, or approximately 801.41 square units.
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Find m/STR.
186
T
m/STR=
112
R
degrees
