Answers
The equation of the normal line to the curve x^2y^2 = 4 at the point (-1,-2) is y = x - 1.
To find the tangent line and normal line to the curve x^2y^2 = 4 at the point (-1,-2), we need to determine the derivative of the curve equation with respect to x and evaluate it at the given point.
First, let's differentiate the equation x^2y^2 = 4 implicitly with respect to x using the chain rule:
2x * (y^2) + 2y * (2xy * dy/dx) = 0
Simplifying the equation, we have:
2xy^2 + 4xy(dy/dx) = 0
Now, let's find the value of dy/dx at the point (-1,-2). Substitute x = -1 and y = -2 into the equation:
2*(-1)(-2)^2 + 4(-1)*(-2)(dy/dx) = 0
Simplifying further:
8 + 8(dy/dx) = 0
8(dy/dx) = -8
dy/dx = -1
We have found the derivative dy/dx at the point (-1,-2), which is -1. This represents the slope of the tangent line to the curve at that point.
Using the point-slope form of a line, we can write the equation of the tangent line as:
y - y₁ = m(x - x₁)
Substituting the values of (-1,-2) and dy/dx = -1 into the equation, we have:
y - (-2) = -1(x - (-1))
y + 2 = -1(x + 1)
y + 2 = -x - 1
y = -x - 3
Therefore, the equation of the tangent line to the curve x^2y^2 = 4 at the point (-1,-2) is y = -x - 3.
To find the normal line, we know that the slope of the normal line is the negative reciprocal of the slope of the tangent line. Therefore, the slope of the normal line is 1.
Using the point-slope form of a line again, we can write the equation of the normal line as:
y - y₁ = m'(x - x₁)
Substituting the values of (-1,-2) and m' = 1 into the equation, we have:
y - (-2) = 1(x - (-1))
y + 2 = x + 1
y = x - 1
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Related Questions
Ai Mi is a teacher and takes home 61 papers to grade over the
weekend. She can grade at a rate of 6 papers per hour. How many
papers would Ai Mi have remaining to grade after working for 8 hours?
Answers
...................................................................
Answer:
Step-by-step explanation:
the answer is 13 remaining papers
A parabola can be drawn given a focus of ... 100pts
Answers
Answer:
The parabola has a vertex at (3, -4), has a p-value of -6 and it opens downwards.
Step-by-step explanation:
The given directrix of the parabola is y = 2, which is a horizontal line.
This means that the parabola is vertical, with a vertical axis of symmetry.
The focus of a parabola is a fixed point located inside the curve. The y-coordinate of the given focus is y = -10. As this is below the directrix, it means that the parabola opens downwards.
The standard form of a vertical parabola is:
[tex]\boxed{(x-h)^2=4p(y-k)}[/tex]
where:
Vertex = (h, k)Focus = (h, k+p)Directrix: y = (k - p)Axis of symmetry: x = hAs the focus is (3, -10), then:
[tex](h, k+p)=(3,-10)[/tex]
[tex]\implies h = 3[/tex]
[tex]\implies k+p=-10[/tex]
As the directrix is y = 2, then:
[tex]k - p=2[/tex]
To find the value of k, sum the equations involved k and p to eliminate p:
[tex]\begin{array}{crcccr}&k &+& p& =& -10\\+&k& -& p& = &2\\\cline{2-6}&2k&&& =& -8\\\cline{2-6}\\\implies &k&&&=&-4\end{array}[/tex]
To find the value of p, substitute the found value of k into one of the equations:
[tex]-4-p=2[/tex]
[tex]p=-4-2[/tex]
[tex]p=-6[/tex]
Therefore, the values of h, k and p are:
h = 3k = -4p = -6The parabola has a vertex at (3, -4), has a p-value of -6 and it opens downwards.
The parabola has a vertex at (3, -4), has a p-value of -6 and it opens downwards.
How to determine the equation and vertex of a parabola?In Mathematics, the standard form of the equation of the directrix lines for any parabola is given by this mathematical equation:
(x - h)² = 4p(y - k).
Where:
h and k are the vertex.p is a point.Since the directrix is horizontal, the axis of symmetry would be vertical. This ultimately implies that, we would have the following parameters;
directrix is y = 2
Focus, (h, k + p) = (3, -10)
Next, we would determine the value of k as follows;
k + p = -10 .......equation 1
k - p = 2 .......equation 2
By solving the equations simultaneously, we have:
2k = -8
k = -4
For the value of p, we have the following from equation 2:
k - p = 2
-4 - p = 2
p = -4 - 2
p = -6
In conclusion, we can logically deduce that the parabola opens downward because the p-value is negative.
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Jaxon's mother spends more than 2 hours cleaning the house. The inequality x> 2 represents the situation. Which
graph represents the inequality?
Answers
Answer:A
Step-by-step explanation:the graph shows the values that are greater than 2
What is the value of x? Triangle ABC. Segment AD bisects angle A. The length of side AB is 28. The length of segment BD is 14. The length of side AC is 25. The length of segment CD is unknown and is labeled x. Enter your answer, as a decimal, in the box. x =
Answers
Answer:
Step-by-step explanation:
To find the value of x, we can use the Angle Bisector Theorem, which states that in a triangle, a line segment that bisects an angle divides the opposite side into segments that are proportional to the other two sides.
In this case, segment AD bisects angle A, so we can set up the following proportion:
BD/DC = AB/AC
Plugging in the given values, we have:
14/DC = 28/25
To solve for DC (segment CD), we can cross-multiply:
28 * DC = 14 * 25
Simplifying further:
DC = (14 * 25) / 28
DC ≈ 12.5
Therefore, the length of segment CD is approximately 12.5.
please answer ASAP I will brainlist
Answers
Answer:
(a) 10, $2088.81
(b) see attached
(c) B. increases at a slower rate
Step-by-step explanation:
Given the cost function g(x) = -1736.7 +1661.4·ln(x) for the average dollar cost of health insurance in year x after 2000, you want the cost in 2010, a graph for the years 2006 to 2015, and a description of the shape of the curve.
(a) Cost in 2010The value of x is years after 2000, so for the year 2010, the value of x is ...
x = 2010 -2000 = 10
Substituting 10 for x in the function will give the cost in 2010. That is ...
g(10) = -1736.7 +1661.4·ln(10) ≈ 2088.81
The cost in 2010 is about $2088.81.
(b) GraphThe second attachment shows a graphing calculator's rendition of the graph. We note it has positive slope everywhere, but does not intersect the lines y=1000 or y=3000. This eliminates choices A, C, and D, leaving choice B for the graph.
(c) ShapeThe logarithm function has positive and decreasing slope. The function here is a scaled and shifted version of the logarithm function, but it still has positive and decreasing slope. That is, ...
the average cost increases at a slower rate as time goes on
<95141404393>
Circumference of circle inscribed or circumscribed polygon
Hint: you will need to find the diameter of the circle, use Pythagorean Theorem)
ind then I out of the 3 problems.
Find the exact circumference of each circle by using the given inscribed or circumscribed polygon.
8 cm
15 cm
Answers
The exact circumferences of the inscribed and circumscribed circles for the given polygons are 8π cm and 15π cm, respectively.
To find the exact circumference of a circle inscribed or circumscribed by a polygon, we can use the Pythagorean theorem to determine the diameter of the circle.
In the case of an inscribed polygon, the diameter of the circle is equal to the diagonal of the polygon. Let's consider the polygon with a diagonal of 8 cm. If we draw a line connecting two non-adjacent vertices of the polygon, we get a diagonal that represents the diameter of the inscribed circle.
Using the Pythagorean theorem, we can find the length of this diagonal. Let's assume the sides of the polygon are a and b. Then the diagonal can be found using the equation: diagonal^2 = a^2 + b^2. Substituting the given values, we have 8^2 = a^2 + b^2. Solving this equation, we find that a^2 + b^2 = 64.
For the circumscribed polygon with a diagonal of 15 cm, the diameter of the circle is equal to the longest side of the polygon. Let's assume the longest side of the polygon is c. Therefore, the diameter of the circumscribed circle is 15 cm.
Once we have determined the diameter of the circle, we can calculate its circumference using the formula C = πd, where C is the circumference and d is the diameter.
For the inscribed circle, the circumference would be C = π(8) = 8π cm.
For the circumscribed circle, the circumference would be C = π(15) = 15π cm.
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The midpoint of AB is M(-4,2). If the coordinates of A are (-7,3), what are the
coordinates of B?
Answers
The midpoint of AB is M(-4,2). If the coordinates of A are (-7,3), and the coordinates of B is (-1, 1).
To find the coordinates of point B, we can use the midpoint formula, which states that the coordinates of the midpoint between two points (A and B) can be found by averaging the corresponding coordinates.
Let's denote the coordinates of point A as (x1, y1) and the coordinates of point B as (x2, y2). The midpoint M is given as (-4, 2).
Using the midpoint formula, we can set up the following equations:
(x1 + x2) / 2 = -4
(y1 + y2) / 2 = 2
Substituting the coordinates of point A (-7, 3), we have:
(-7 + x2) / 2 = -4
(3 + y2) / 2 = 2
Simplifying the equations:
-7 + x2 = -8
3 + y2 = 4
Solving for x2 and y2:
x2 = -8 + 7 = -1
y2 = 4 - 3 = 1
Therefore, the coordinates of point B are (-1, 1).
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Write a linear equation for the following table.
x = number
y = cost
0
15
35
55
75
95
y =
4
8
12
16
X +
Answers
Answer:
[tex]m = \frac{35 - 15}{4 - 0} = \frac{20}{4} = 5[/tex]
[tex]y = 5x + 15[/tex]
Given that p(x)=2(5−x)2+1 , what is the value of p(-4)? Responses
Answers
Answer:
37
Step-by-step explanation:
x=-4
=2(5-(-4)2+1
=2(5+4)2+1
=2(9)2+1
=18(2)+1
=36+1
=37
14x^(2n+1)+7x^(n+3)-21^(n+2)
100 points will be awarded
Answers
Answer:
Step-by-step explanation:
The given expression is: 14x^(2n+1) + 7x^(n+3) - 21^(n+2)
Unfortunately, it seems there is a missing exponent for the term "21" in the expression. Please provide the correct exponent for 21, and I'll be happy to help you further simplify the expression.
On a line graph, time is usually represented on the vertical axis.
O True
O False
--
Answers
labor-hours and its standard cost card per unit is as follows:
Direct material: $ pounds at $11.00 per pound
Direct labor: 3 hours at $12 per hour
Variable overhead: 3 hours at $7 per hour
Total standard variable cost per unit
The company also established the following cost formulas for its selling expenses:
sales salaries and commissions
shipping expenses
Fixed Cost per
Month
$ 280,000
$ 260,000
$ 55.00
36.00
$112.00
Variable
Cost per
Unit Sold
$ 20.00
$ 11.00
The planning budget for March was based on producing and selling 21,000 units. However, during March the company
actually produced and sold 26.600 units and incurred the following costs:
a Purchased 154.000 pounds of raw materials at a cost of $9.50 per pound. All of this material was used in production.
b. Direct laborers worked 63,000 hours at a rate of $13.00 per hour
e Total variable manufacturing overhead for the month was $510,930
d Total advertising sales salaries and commissions, and shipping expenses were $286,000, $495,000, and $195,000,
respectively
6 What direct labor cost would be included in the company's flexible budget for March?
Answers
The direct labor cost included in the Preble Company's flexible budget for March is $819,000.
How to compute Preble Company's direct labor cost?To find the direct labor cost included in the company's flexible budget for March, we shall estimate the actual direct labor cost incurred during the period.
Given:
Actual production and sales =n26,600 units
Actual direct labor rate = $13.00 per hour
Actual direct labor hours worked = 63,000 hours
Direct labor cost = Actual direct labor rate × Actual direct labor hours worked
Direct labor cost = $13.00/hour × 63,000 hours
Direct labor cost = $819,000
Hence, the direct labor cost included in the company's flexible budget for March would be $819,000.
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Find the length of X
Answers
Answer:
We have similar triangles.
7/3.5 = 5/x
2 = 5/x, so x = 2.5
On a coordinate plane, a dashed straight line has a positive slope and goes through (negative 3, negative 3) and (3, 1). Everything below and to the right of the line is shaded.
Which linear inequality is represented by the graph?
y > Two-thirdsx – 2
y < Two-thirdsx + 2
y > Two-thirdsx + 1
y < Two-thirdsx – 1
Answers
Answer:
y > Two-thirds x + 1 /c
Let S be a set that contains atleast two different elements. Let R be the relation on P(S),
thesetofallsubsetsofS,definedby(X,Y)∈RifandonlyifX∩Y =φ.
(i) Determine whether R is reflexive, symmetric, antisymmetric, or transitive.
(ii) Why is it important that S has atleast 2 different elements?
(iii) Would any of the answers change in (i) if S was empty or had only one element?
Answers
(i) R is not reflexive, symmetric, vacuously antisymmetric, and not transitive.
(ii) It is important that S has at least two different elements because if S had only one element, then the power set P(S) would contain only two sets: the empty set and the set S itself.
(iii) If S was empty, there would be no elements in S, and hence the power set P(S) would only contain the empty set. The answers regarding reflexivity, symmetry, antisymmetry, and transitivity would all change, and the relation R would not possess any of these properties. If S had only one element, the power set P(S) would contain two sets: the empty set and the singleton set containing the one element of S. The relation R would be reflexive, symmetric, antisymmetric, and transitive, as it satisfies all these properties by definition.
(i)Let's analyze the properties of relation R on P(S):
Reflexive: A relation R is reflexive if for every element x in the set S, (x, x) belongs to R. In this case, since the intersection of any set with itself is never empty, the relation R is not reflexive.
Symmetric: A relation R is symmetric if for every pair (x, y) in R, (y, x) also belongs to R. In this case, since the intersection of two sets is commutative, if (X, Y) belongs to R, then (Y, X) also belongs to R. Thus, the relation R is symmetric.
Antisymmetric: A relation R is antisymmetric if for every distinct pair (x, y) in R, (y, x) does not belong to R. Since the relation R is defined by the condition X∩Y = φ, there are no distinct pairs (X, Y) that satisfy the condition. Therefore, the relation R is vacuously antisymmetric.
Transitive: A relation R is transitive if for every three sets X, Y, and Z such that (X, Y) belongs to R and (Y, Z) belongs to R, then (X, Z) also belongs to R. In this case, if X∩Y = φ and Y∩Z = φ, it does not imply that X∩Z = φ. Hence, the relation R is not transitive.
(ii) In this scenario, the relation R would not have any pairs (X, Y) where X and Y are non-empty subsets of S, because the intersection of any two non-empty subsets would never be empty. Thus, the relation R would be trivial and uninteresting.
(iii) In this case, since there are no non-empty subsets in P(S), the relation R would not have any pairs (X, Y) where X and Y are non-empty subsets of S.
In this case, the relation R would have only one possible pair (X, Y) with X and Y being the empty set, and their intersection would indeed be empty.
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I need help please!!
Answers
Answer:
(r q)(-3) = -3
(q r)(-3) = -3
Step-by-step explanation:
let x = 1
q(1) = -1 +2 = 1
r(1) = 1² = 1
(r q)(-3) = ?
(1×1)(-3) = -3
(q r)(-3) = ?
(1×1)(-3) = -3
What is the prime factorization of 625?
Answers
Answer:
5⁴
Step-by-step explanation:
625= 5×5×5×5
help me please.. please
Answers
Step-by-step explanation:
Parallel to the x-axis means it is just a horizontal line with the value being the y-coordinate of the point:
y = -2
Answer:
y=-2
and m=0 must be your answer
Step-by-step explanation:
as line is parallel to x axis its slope will be zero as it does not have any definite x coordinate
so
equation of line is y-y'=m(x-x')
so m=0 m is slope
y'=-2 and x'=4
so by substituting the values
y+2=0
so y=-2
and m=0 is your answer
. Julio compró ropa por un costo de $5000, por la cual dejó el 20% de apartado. ¿Con cuánto dinero termina de pagar la ropa?
Answers
Julio bought clothes for $5000 with a 20% down payment, which amounts to $1000. Hence, he ends up paying $4000 for the clothes.
Julio's clothing purchase involved a total cost of $5000. To secure the purchase, he made a down payment of 20% of the total cost. To calculate the down payment, we multiply the total cost by the down payment percentage:
Down payment = 20% * $5000
Down payment = 0.20 * $5000
Down payment = $1000
The down payment amount is $1000. To determine the final amount that Julio ends up paying for the clothes, we need to subtract the down payment from the total cost:
Total cost - Down payment = $5000 - $1000
Total cost - Down payment = $4000
Therefore, Julio ends up paying $4000 for the clothes.
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The question probable may be:
Julio bought clothes for a cost of $5000, for which he left a 20% down payment. How much money does he end up paying for the clothes?
f.) Use the regression line to calculate the best predicted value for x = 32. (Round to two decimal places)
y =
Answers
The best predicted value for x = 32 is -38333.33
Estimating the predicted value at x = 32 for the regression lineFrom the question, we have the following parameters that can be used in our computation:
The scatter plot (see attachment)
We have the following points from the line of best fit
(3, 10000) and (0, 15000)
The linear equation is represented as
y = mx + c
Where
c = y when x = 0
So, we have
y = mx + 15000
Using the other point, we have
10000 = 3m + 15000
So, we have
3m = -5000
Divide by 3
m = -5000/3
So, the equation is y = -5000/3x + 15000
When x = 32, we have
y = -5000/3 * 32 + 15000
Evaluate
y = -38333.33
Hence, the best predicted value for x = 32 is -38333.33
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Help excel college student
EOP511
Answers
The total petty cash expenditures would be =$130.84
How to calculate the petty cash expenditures?To calculate the petty cash expenditures, the following is added up as follows;
The cost for stamp = $12.50
The cost for coffee supplies = $25.19
The cost for pizza delivery = $15.50
The cost for white board markers = $20.00
The cost for sympathy greeting card= $5.78
The cost of flowers for Jean's retirement farewell = $39.87
The cost of courier = $12.00
Therefore the total petty cash expenditures would be= 12.50+25.19+15.50+20+5.78+39.87+12= $130.84
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Jerry tries to find the measure of ABC His work is shown. Which statement describes Jerry's error? The picture shows a circle with a center O. Two chords, BA and BC are drawn from the same internal point, B. The angle of A is 135 degrees and C is 119 degrees.
Answers
Jerry's error is that he incorrectly assumes that the measure of angle ABC is equal to the sum of the measures of angles A and C. Hence Statement B is answer.
This is not true because angle ABC is an inscribed angle, which means its measure is half the measure of the intercepted arc.
In this case, the intercepted arc is AC. Since angle A measures 135 degrees and angle C measures 119 degrees, the sum of their measures is 254 degrees. However, angle ABC is not equal to 254 degrees.
To find the measure of angle ABC, we need to find the measure of arc AC. Since arcs A and C are equal in measure, we can find the measure of arc AC by subtracting angle A's measure (135 degrees) from the full circle measure (360 degrees).
Therefore, the measure of arc AC is 360 - 135 = 225 degrees. Since angle ABC is an inscribed angle, its measure is half the measure of arc AC, which is 225/2 = 112.5 degrees.
Hence Statement B is answer.
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A rectangular piece of metal is 5 in longer than it is wide. Squares with sides 1 in lòng are cut from the four corners
and the flaps are folded upward to form an open box. If the volume of the box is 234 in³, what were the original
dimensions of the piece of metal?
Answers
The original dimensions of the piece of metal were 15 inches by 20 inches.
To solve this problem, we can use the given information to set up an equation. Let's assume that the width of the rectangular piece of metal is x inches. According to the problem, the length of the piece of metal is 5 inches longer than its width, so the length would be (x+5) inches.
When squares with sides 1 inch long are cut from the four corners, the width and length of the resulting box will be reduced by 2 inches each. Therefore, the width of the box will be (x-2) inches and the length will be ((x+5)-2) inches, which simplifies to (x+3) inches.
The height of the box will be 1 inch since the flaps are folded upward.
Now, let's calculate the volume of the box using the formula Volume = length * width * height.
Substituting the values, we have:
234 = (x+3)(x-2)(1)
Simplifying the equation, we get:
234 = x^2 + x - 6
Rearranging the equation, we have:
x^2 + x - 240 = 0
Now, we can solve this quadratic equation either by factoring or by using the quadratic formula. Let's use factoring to find the values of x.
Factoring the equation, we have:
(x+16)(x-15) = 0
Setting each factor equal to zero, we get:
x+16 = 0 or x-15 = 0
Solving for x, we have:
x = -16 or x = 15
Since the width cannot be negative, we take x = 15 as the valid solution.
Therefore, the original dimensions of the piece of metal were 15 inches in width and (15+5) = 20 inches in length.
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(2, 5); (3, 6); (4, 11); (5, 14); (6, 15). The following linear function is fitted to the data: y = 2x + 2. Plot the residuals.
Answers
For the data point (2, 5), the residual is 5 - 6 = -1. For the data point (3, 6), the residual is 6 - 8 = -2. For the data point (4, 11), the residual is 11 - 10 = 1. For the data point (5, 14), the residual is 14 - 12 = 2. For the data point (6, 15), the residual is 15 - 14 = 1.
To plot the residuals for the given linear function, we need to calculate the difference between the actual y-values of the data points and the corresponding predicted y-values based on the linear function.
Let's start by calculating the predicted y-values using the linear function y = 2x + 2 for each x-value in the dataset:
For x = 2, the predicted y-value is 2(2) + 2 = 6.
For x = 3, the predicted y-value is 2(3) + 2 = 8.
For x = 4, the predicted y-value is 2(4) + 2 = 10.
For x = 5, the predicted y-value is 2(5) + 2 = 12.
For x = 6, the predicted y-value is 2(6) + 2 = 14.
Now we can calculate the residuals by subtracting the predicted y-values from the actual y-values.
To visualize these residuals, we can create a scatter plot with the x-values on the horizontal axis and the residuals on the vertical axis. Each data point will represent the x-value from the dataset and its corresponding residual. The residuals can be either positive or negative, so we should include both sides of the vertical axis.
By plotting the residuals, we can observe the deviations between the actual y-values and the predicted y-values based on the linear function. This plot helps us assess how well the linear function fits the data and identify any patterns or trends in the residuals.
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If P= (4,2) Find: RX=3 (P)
Answers
Answer: 2,2
Step-by-step explanation:
trust me
Using exactly nine bills, how can you make change for $55 that will NOT make change for a twenty dollar bill?
a.
1 twenty, 3 tens, 5 singles
c.
2 twenties, 1 ten, 6 singles
b.
4 tens, 2 fives, 5 singles
d.
2 twenties, 2 fives, 5 singles
Answers
The correct combination is option a) 1 twenty, 3 tens, and 5 singles, which allows us to make change for $55 without making change for a twenty dollar bill.
The correct answer is a) 1 twenty, 3 tens, 5 singles.
To make change for $55 using exactly nine bills without making change for a twenty dollar bill, we need to avoid using any combination that includes a twenty dollar bill.
Option a) includes 1 twenty, 3 tens, and 5 singles. The total value of these bills is 20 + 3(10) + 5(1) = $55. This combination allows us to make the exact change for $55 without including a twenty dollar bill.
Option b) includes 4 tens, 2 fives, and 5 singles. The total value of these bills is 4(10) + 2(5) + 5(1) = $55. Although this combination also makes the exact change for $55, it includes four tens, which can be exchanged for a twenty dollar bill.
Option c) includes 2 twenties, 1 ten, and 6 singles. The total value of these bills is 2(20) + 10 + 6(1) = $57. This combination exceeds $55 and also includes two twenty dollar bills, making change for a twenty dollar bill.
Option d) includes 2 twenties, 2 fives, and 5 singles. The total value of these bills is 2(20) + 2(5) + 5(1) = $55. However, this combination includes two twenty dollar bills, making change for a twenty dollar bill.
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Find the value of x.
*
20%
x=
104
degrees
Answers
Answer:
x=4 explanation: I got it right
Similar Triangles
Determine whether the triangles are similar. If so, write a similarity statement. If not, what would be sufficient to
prove the triangles similar? Explain your reasoning.
I need help on number 1 and 2
Answers
The equivalent ratio of the corresponding sides and the triangle proportionality theorem indicates that the similar triangles are;
1. ΔAJK ~ ΔSWY according to the SAS similarity postulate
2. ΔLMN ~ ΔLPQ according to the AA similarity postulate
3. ΔPQN ~ ΔLMN
LM = 12, QP = 8
4. ΔLMK~ΔLNJ
NL = 21, ML = 14
What are similar triangles?
Similar triangles are triangles that have the same shape but may have different sizes.
1. The ratio of corresponding sides between the two triangles circumscribing the congruent included angle are;
24/16 = 3/2
18/12 = 3/2
The ratio of each of the two sides in the triangle ΔAJK to the corresponding sides in the triangle ΔSWY are equivalent and the included angle, therefore, the triangles ΔAJK and ΔSWY are similar according to the SAS similarity rule.
2. The ratio of the corresponding sides in each of the triangles are;
MN/LN = 8/10 = 4/5
PQ/LQ = 12/(10 + 5) = 12/15 = 4/5
The triangle proportionality theorem indicates that the side MN and PQ are parallel, therefore, the angles ∠LMN ≅ ∠LPQ and ∠LNM ≅ ∠LQP, which indicates that the triangles ΔLMN and ΔLPQ are similar according to the Angle-Angle AA similarity rule
3. The alternate interior angles theorem indicates;
Angles ∠PQN ≅ ∠LMN and ∠MLN ≅ ∠NPQ, therefore;
ΔPQN ~ ΔLMN by the AA similarity postulate
LM/QP = (x + 3)/(x - 1) = 18/12
12·x + 36 = 18·x - 18
18·x - 12·x = 36 + 18 = 54
6·x = 54
x = 54/6 = 9
LM = 9 + 3 = 12
QP = x - 1
QP = 9 - 1 = 8
4. The similar triangles are; ΔLMK and ΔLNJ
ΔLMK ~ ΔLNJ by AA similarity postulate
ML/NL = (6·x + 2)/(6·x + 2 + (x + 5)) = (6·x + 2)/((7·x + 7)
ML/NL = LK/LJ = (24 - 8)/24
(24 - 8)/24 = (6·x + 2)/((7·x + 7)
16/24 = (6·x + 2)/(7·x + 7)
16 × (7·x + 7) = 24 × (6·x + 2)
112·x + 112 = 144·x + 48
144·x - 112·x = 32·x = 112 - 48 = 64
x = 64/32 = 2
ML = 6 × 2 + 2 = 14
NL = 7 × 2 + 7 = 21
MN = 2 + 5 = 7
Learn more on similar triangles here: https://brainly.com/question/2644832
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What kind of growth model (pattern) is shown in the table?
x
y
1
5
2
25
3
125
4
625
5
3,125
square root
linear
exponential
quadratic
Answers
Answer:
Option C is correct.
The kind of growth model is shown in the table is exponential
Step-by-step explanation:
Exponential growth function is in the form of : ......[1]; where a is the initial value and b> 0.
Consider any two point from the table:
(1 , 5) and ( 2 , 25)
Substitute these in the equation [1] we get;
......[2]
......[3]
Divide equation [3] by [2] we have;
Simplify:
Now substitute this value in equation [2] we get;
Divide both sides by 5 we get;
Simplify:
1=a or a = 1
Therefore, the table shown the exponential growth function y=5^x
The exponential growth model y = Ae^rt can be used to calculate the future population of a city. In this model, A is the current population, r is the rate of growth, and y is the future population for a specific time, t, in years.
A certain city's population has a growth rate of r = 0.08. Approximately how long will it take the city's population to grow from 250,000 to 675,000?
NEED ASAP
Answers
Step-by-step explanation:
in the formula
y = Ae^rt
y is 675,000
A is 250,000
r is 0.08
to get the value of t
y = Ae^rt
y/A = e^rt
ln(y/A) = rt
[ln(y/A)]/r = t
y = Ae^(rt)
In this case, the initial population (A) is 250,000, and the future population (y) is 675,000. The growth rate (r) is given as 0.08. We need to solve for time (t).
675,000 = 250,000 * e^(0.08t)
To solve for t, we can take the natural logarithm (ln) of both sides:
ln(675,000) = ln(250,000 * e^(0.08t))
ln(675,000) = ln(250,000) + ln(e^(0.08t))
ln(675,000) = ln(250,000) + 0.08t
Now, we can isolate t by subtracting ln(250,000) and dividing by 0.08:
0.08t = ln(675,000) - ln(250,000)
t = (ln(675,000) - ln(250,000)) / 0.08
Calculating this value will give us the approximate time it takes for the population to grow from 250,000 to 675,000.
