Question 1[16 marks] Consider the following optimisation problem max f(x, y) = t √ x y, subject to tx2 + y ≤ 5 x ≥ 0, y ≥ 0.
a) Solve the problem for t = 1.
b) State and explain the content of the envelope theorem.
c) What is the marginal effect on the solution if the constant t is increased?
Answers
a) The required value is 2.7778.
b) if a parameter changes slightly, the change in the optimal value can be estimated using the first-order approximation.
c) an increase in t will lead to a higher value of f(x, y).
a) Optimization problem: The optimization problem is shown below:
max f(x, y) = t √ x y, subject to tx² + y ≤ 5x ≥ 0y ≥ 0
Solving the problem for t = 1,t = 1f(x,y) = √xytx² + y ≤ 5x ≥ 0y ≥ 0.
The Lagrangian function for this problem is:
L(x, y, λ) = t √ xy + λ(5 - tx² - y)
We set the partial derivative of L with respect to x to zero:
∂L/∂x = t(0.5√y)/√x + (-2λtx) = 0
We then obtain:
(1) 0.5t√y/√x = 2λtx
If we set the partial derivative of L with respect to y to zero, we obtain:
(2) 0.5t√x/√y + λ(-1) = 0
Multiplying both sides by (-1), we obtain:
(3) -0.5t√x/√y = λ
We set the partial derivative of L with respect to λ to zero, we obtain:
(4) 5 - tx² - y = 0
Substituting Equation (3) into Equation (1), we obtain:
(5) 0.5t√y/√x = -2(5 - tx² - y)x
Substituting Equation (5) into Equation (4), we obtain:
(6) 5 - tx² - 2x²(5 - tx² - y)² = 0
After expanding Equation (6), we obtain a fourth-order equation in y. Solving this equation leads to:(7)
y = 5 - tx²/3
We then substitute y into Equation (3) to obtain:
x = 5/2t²
From Equation (7), we obtain: y = 5 - tx²/3=5-5/3*2.5=2.7778
The required value is 2.7778.
(b) Envelope theorem
According to the Envelope Theorem, the marginal effect of a parameter on an optimal solution is equal to the partial derivative of the optimal value with respect to that parameter. This means that if a parameter changes slightly, the change in the optimal value can be estimated using the first-order approximation.
(c) Increasing the constant tIf we increase the constant t, the optimal x and y will also increase. This is because an increase in t will lead to a higher value of f(x, y).
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Related Questions
can someone please help me? ive been stuck on this for 5 hours
In order to keep the Bermuda Triangle on the Flat Earth from toppling over, it was balanced on the hole which leads to the interior of the hollow Earth. If the vertices of the Bermuda Triangle are given by (1,3) , (2,7) , and (6,2). In order to find the hole we will need to find the intersection of at least 2 lines. Name 2 points through which one of those lines will pass (in the form (x,y),(x,y))
Answers
The following are the two lines: y = 4x - 1. y = -1.25x + 10.5.
Their point of intersection is: (2.19, 7.76).
The two vertices through which the line y = 4x - 1 pass are as follows:
(2.19, 7.76) and (3,11).
The slope is:
m = (7 - 3)/(2 - 1) = 4.
Hence:
y = 4x + b.
When x = 1, y = 3, the intercept b is calculated as follows:
3 = 4 + b
b = -1.
The first line is then: y = 4x - 1.
The points for the second line are (2,7) and (6,2).
As a result, the slope is m = (2 - 7)/(6 - 2) = -5/4 = -1.25.
The solution is y = -1.25x + b.
When x = 2, y = 7, therefore the intercept b is as follows:
7 = -1.25(2) + b
b = 10.5.
As a result, the second line is:
y = -1.25x + 10.5.
The intersection's x-coordinate is as follows:
4x - 1 = -1.25x + 10.5
5.25x = 11.5
x = 11.5/5.25
x = 2.19.
The intersection's y-coordinate is:
y = -1.25(2.19) + 10.5 = 7.76.
Another position where the line y = 4x - 1 intersects is:
y = 4(3) - 1 = 11.
As a result, point (3,11).
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What is the other endpoint of the segment with midpoint 23 and endpoint 27
Answers
Answer:
19
Step-by-step explanation:
Difference between one end point and mid point
= 27 - 23 = 4
Other endpoint = 23 - 4 = 19
what is the quotient of the expression
\( \frac{21a {}^{3} b - 14ab {}^{2} + 7ab}{7ab} \)
Answers
4/6 simplfy it can someone help
Answers
4/6=2/3 (divide both numerator and denominator by 2 (gcd)).
Hope this helps.
2/3
Hope this helps!! Good luck! :)
The width is to be 17 feet less than 3 times the height. Find the width and the height of the carpenter expects to use 30 feet of lumber to make it.
Answers
The height of the carpenter's creation is 8 feet and the width is 7 feet.
To solve this problem, we can use two equations. The first equation is based on the relationship between the width and height:
width = 3(height) - 17
The second equation is based on the amount of lumber the carpenter has available:
2(width) + 2(height) = 30
We can substitute the first equation into the second equation to solve for height:
2(3(height) - 17) + 2(height) = 30
6(height) - 34 + 2(height) = 30
8(height) = 64
height = 8
Using the first equation, we can solve for the width:
width = 3(8) - 17 = 24 - 17 = 7
Therefore, the height is 8 feet and the width is 7 feet.
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A. suppose you have 40 college freshman who have volunteered to be part of this study. design a completely randomized experiment that tests the hypothesis that hearing older students talk about overcoming challenges improves academic performance. be sure your design addresses how randomization will be incorporated.
b. suppose you have reason to believe that students with a history of strong academic performance respond differently than those with a history of modest performance. outline a randomized block design using the same subjects that will address this difference.
Answers
The completely randomized experiment that tests the hypothesis is Illustrated below.
How to illustrate the experiment?a. Freshmen of Group 1 are exposed to ‘hearing older students talk about overcoming challenges’ for a specified period. This group forms the treatment group.
Freshmen of Group 2 are not exposed to ‘hearing older students talk about overcoming challenges. This group forms the control group.
At the end of the specified period, administer the same test to both groups to evaluate their academic performance and their score would form the response variable.
Test the hypotheses: Null: H0: µ1 = µ2 Vs Alternative: HA: µ1 > µ2, where µ1 and µ2 are the population means corresponding to Group 1 and Group 2 respectively, employing 2-sample independent t-test.
If test rejects the null hypothesis, conclude that hearing older students talk about overcoming challenges improves academic performance
b. Here, the blocking factor would be: history of academic performance. Based on this factor, classify the 40 freshman students into 2 or 3 groups – each group forms a block.
From each block, assign randomly an equal number of students to treatment and control groups.
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26. A square garden of side 10 m has its corners filled with flowers. The corners are
squares of side 1m. What is the
i) Area of garden
ii) Area of each flower bed
Hi) Area of the remaining garden
iv) Length of the fence around the garden
v) Cost of planting grass in the remaining figure at the rate of 20 per sqm
Answers
Answer:
wooohohh what is that it's not a question lol i can't find the question to answer youu
Prove using rules of inference 1. If the band could not play rock music or the refreshments were not delivered on time, then the New Year's party would have been canceled and Alicia would have been angry. If the party were canceled, then refunds would have had to be made. No refunds were made. Therefore the band could play rock music. 2. If you are not in the tennis tournament, you will not meet Ed. If you aren't in the tennis tournament or if you aren't in the play, you won't meet Kelly. You meet Kelly or you meet Ed. It is false that you are in the tennis tournament and in the play. Therefore, you are in the tennis tournament.
Answers
The main answer for the first argument is that we cannot prove that the band could play rock music based on the given premises and rules of inference.
1. Let's assign the following propositions:
- P: The band could play rock music.
- Q: The refreshments were delivered on time.
- R: The New Year's party was canceled.
- S: Alicia was angry.
- T: Refunds were made.
2. The given premises can be expressed as:
(¬P ∨ ¬Q) → (R ∧ S)
R → T
3. To prove that the band could play rock music (P), we need to derive it using valid rules of inference.
4. Using the premises, we can apply the rule of modus tollens to the second premise:
R → T (Premise)
Therefore, ¬R.
5. Next, we can use disjunctive syllogism on the first premise:
(¬P ∨ ¬Q) → (R ∧ S) (Premise)
¬R (From step 4)
Therefore, ¬(¬P ∨ ¬Q).
6. Applying De Morgan's law to step 5, we get:
¬(¬P ∨ ¬Q) ≡ (P ∧ Q)
7. Therefore, we can conclude that the band could play rock music (P) based on the premises and rules of inference.
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I need help on this ASAP the equation is shown there
Answers
Answer:
slope -3, y intercept 15
Step-by-step explanation:
Slope intercept for of equation is:
y=mx+b
Given equation is in form of
d=15-3v
From this we can conclude the slope is -3
y-intercept is 15
what is 2 3/5 as a decimal
Answers
Answer: 2.6
Step-by-step explanation:
1: Split the number into its whole number component and it fractional component.
2 3/5=2+3/5.
2: For the denominator, recognize that 5x20=100.
Multiple=20.
3: Multiply both the numerator and denominator by the multiple 20.
3x20, 5x20.
4: Simplify.
60/100.
5: Simplify.
0.6
6: Combine the whole number component with the decimal.
2.6✅
without actually solving the given differential equation, find the minimum radius of convergence r of power series solutions about the ordinary point x = 1. (x^2 - 2x + 17)y"+ xy' -4y = 0
Answers
Power series solutions have a minimum radius of convergence of R of 10.0498 around the normal point x = 0 and 10 units around the normal point x=1.
What is a differential equation?A differential equation in mathematics is an equation that connects the derivatives of one or more unknown functions.
Applications often involve functions that reflect physical quantities, derivatives that depict the rates at which those values change, and a differential equation that establishes a connection between the three.
The given equation: \(\left(x^2-2 x+26\right) y^{\prime \prime}+x y^{\prime}-4 y=0\)
It is necessary to determine the power series solutions' minimal radius of convergence R around the typical points x = 0 and x = 1.
The separation between the ordinary point and the differential equation's singularity is now the minimal radius of convergence.
The polynomial's root, which is connected to the second derivative, is the singularity point.
The singularity points will be determined as follows:
\(\begin{aligned}& x^2-2 x+26=0 \\& x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} \\& x=\frac{2 \pm \sqrt{(-2)^2-4 \times 1 \times 26}}{2} \\& x=1 \pm \sqrt{-100} \\& x=1 \pm 10 i\end{aligned}\)
In this case, x1 = 1+10i and x2 = 1-10i are the singularity sites.
The ordinary points at this time are z1 = 0+01 and z2 = 1+0i.
One can compute the minimum radius of convergence using the formula:
\(\begin{aligned}& r_1=\left|z_1-x_1\right| \\& =|0+0 i-1-10 i| \\& =\sqrt{101} \\& =10.0498 \\& r_2=\left|z_2-x_1\right| \\& =\sqrt{100} \\& =10\end{aligned}\)
Therefore, power series solutions have a minimum radius of convergence of R of 10.0498 around the normal point x = 0 and 10 units around the normal point x = 1.
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The cost to make each T-shirt is $10. You estimate that you will
sell 50 shirts. If you want to make a profit of at least $250, what
price will you charge for these T-shirts? Show your solution in two
different ways.
Answers
The price per T-shirt should be at least $15 to achieve a profit of $250.
To calculate the price per T-shirt that will yield a profit of at least $250, we need to consider the cost of production, the desired profit, and the number of shirts to be sold.
Given that the cost to make each T-shirt is $10, and we want to sell 50 shirts, the total cost of production would be 10 * 50 = $500.
Now, let's calculate the minimum revenue needed to achieve a profit of $250. We add the desired profit to the total cost of production: $500 + $250 = $750.
Finally, to determine the price per T-shirt, we divide the total revenue by the number of shirts: $750 ÷ 50 = $15.
Therefore, to make a profit of at least $250, the price per T-shirt should be set at $15.
By selling each T-shirt for $15, the total revenue would be $15 * 50 = $750. From this revenue, we subtract the total production cost of $500 to calculate the profit, which amounts to $750 - $500 = $250. Thus, by charging $15 per T-shirt, the desired profit of $250 is achieved.
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A tank of water in the shape of a cone is being filled with water at a rate of 12 m/sec. The base radius of the tank is 26 meters, and the height of the tank is 18 meters. At what rate is the depth of
Answers
The depth of the water in the cone-shaped tank is increasing at a rate of approximately 1.385 meters per second.
To determine the rate at which the depth of the water is changing, we can use related rates. Let's denote the depth of the water as h(t), where t represents time. We are given that dh/dt (the rate of change of h with respect to time) is 12 m/sec, and we want to find dh/dt when h = 18 meters.
To solve this problem, we can use the volume formula for a cone, which is V = (1/3)πr^2h, where r is the base radius and h is the depth of the water. We can differentiate this equation with respect to time t, keeping in mind that r is a constant (since the base radius does not change).
By differentiating the volume formula with respect to t, we get dV/dt = (1/3)πr^2(dh/dt). Now we can substitute the given values: dV/dt = 12 m/sec, r = 26 meters, and h = 18 meters.
Solving for dh/dt, we have (1/3)π(26^2) (dh/dt) = 12 m/sec. Rearranging this equation and solving for dh/dt, we find that dh/dt is approximately 1.385 meters per second. Therefore, the depth of the water in the tank is increasing at a rate of about 1.385 meters per second.
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What’s the value of x
Answers
Answer:
-10
Step-by-step explanation:
-5x - 1 = -10x - 51
Add 10x to both sides.
5x - 1 = -51
Add 1 to both sides.
5x = -50
Divide both sides by 5.
x = -10
I hope this helped! :)
solve for the roots in simplest form using the quadratic formula 4x^2-12x=-17
Answers
9514 1404 393
Answer:
x = 3/2 ± i√2
Step-by-step explanation:
For quadratic ax² +bx +c = 0, the roots are given by the formula ...
\(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\)
When we put your equation into standard form:
4x^2 -12x +17 = 0
we see the relevant formula values are ...
a = 4, b = -12, c = 17
Then the roots are ...
\(x=\dfrac{-(-12)\pm\sqrt{(-12)^2-4(4)(17)}}{2(4)}=\dfrac{12\pm\sqrt{-128}}{8}\\\\x=\dfrac{3\pm2\sqrt{2}i}{2}=\boxed{\dfrac{3}{2}\pm i\sqrt{2}}\)
You flip a 3 coins 50 times, and flipping 3 tails occurs 6 times, flipping 3 heads occurs 7 times.
A. What is the theoretical probability that you flip 3 heads
B.What is the theoretical probability that you flip leas than 3 heads
Answers
Hello!
Answer:
A. The theoretical probability of flipping 3 heads on one flip of three coins is (1/2) * (1/2) * (1/2) = 1/8. Since you flipped three heads 7 times out of 50, the experimental probability of flipping 3 heads is 7/50.
B. To calculate the theoretical probability of flipping less than 3 heads, we can calculate the probability of flipping 0, 1, or 2 heads and add them together. The probability of flipping 0 heads is (1/2) * (1/2) * (1/2) = 1/8, since all three coins must come up tails. The probability of flipping 1 head is (1/2) * (1/2) * (1/2) * 3 = 3/8, since there are three ways to get one head (HTT, THT, TTH). The probability of flipping 2 heads is (1/2) * (1/2) * (1/2) * 3 = 3/8, since there are three ways to get two heads (HHT, HTH, THH). Therefore, the theoretical probability of flipping less than 3 heads is:
1/8 + 3/8 + 3/8 = 7/8.
So the theoretical probability of flipping less than 3 heads is 7/8.
PLEASE HELP! DUE TODAY! TY!
In the figure, line segment AB is tangent to the circle at point A. Find the length of line segment AB.
Work:
AB =
Answers
Answer: 12 inches
Step-by-step explanation:
assume that has designed its servers so there is probability that the number of amazon alexa app downloads in a day exceeds the servers' capacity and more servers have to be brought online. how many amazon alexa app downloads per day are 's servers designed to handle (to the nearest whole number)?
Answers
As per the probability of downloads the capacity of Goo-gle's servers is designed to handle Ama-zon Ale-xa app downloads per day is equal to 4800.
Probability that there are 2000 or fewer downloads of Ama-zon Ale-xa in a day,
Standardize the value using the z-score
Probability using a standard normal distribution table,
z = (2000 - 2800) / 860
= -0.9302
Using a standard normal distribution table,
Probability of a standard normal variable being less than or equal to -0.9302 is 0.1762.
Probability of 2000 or fewer downloads of Ama-zon Al-exa in a day is 0.1762.
Probability that there are between 1500 and 2500 downloads of Ama-zon Ale-xa in a day,
Standardize the values using the z-score
Probability using a standard normal distribution table,
z₁= (1500 - 2800) / 860
= -1.5279
z₂ = (2500 - 2800) / 860
= -0.3488
Using a standard normal distribution table,
Probability of a standard normal variable being less than or equal to -0.3488 is 0.3632.
Probability of a standard normal variable being less than or equal to -1.5279 is 0.0630.
Probability of between 1500 and 2500 downloads of Ama-zon Ale-xa in a day is,
0.3632 - 0.0630
= 0.3002
Probability that there are more than 3000 downloads of Ama-zon Ale-xa in a day.
Standardize the value using the z-score
Probability using a standard normal distribution table,
z = (3000 - 2800) / 860
= 0.2326
Using a standard normal distribution table,
Probability of a standard normal variable being greater than 0.2326 is 0.4090.
Probability of more than 3000 downloads of Ama-zon Ale-xa in a day is 0.4090.
Capacity of Goo-gle's servers,
Probability of the number of Ama-zon Ale-xa app downloads in a day exceeding the capacity is 0.01.
Let's assume that the number of downloads per day follows a normal distribution ,
Mean =2800
Standard deviation = 860.
Let C be the capacity of Goo-gle's servers.
Let X is the number of Ama-zon Ale-xa app downloads in a day.
P(X > C) = 0.01
Using the standard normal distribution,
P(X > C)
= P((X - μ) / σ > (C - μ) / σ)
= P(Z > (C - 2800) / 860)
= 0.01
where Z is a standard normal variable.
Using a standard normal distribution table,
z-score corresponding to a probability of 0.01 is 2.3263.
This implies
(C - 2800) / 860 = 2.3263
Solving for C, we get,
⇒C = 4800.618
⇒C = 4800(Rounding to the nearest whole number)
Therefore, , the capacity of Goo-gle's servers is designed to handle Ama-zon Ale-xa app downloads per day as per the probability is equal to 4800.
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The above question is incomplete, the complete question is:
Ale-xa is the popular virtual assistant developed by Ama-zon. Ale-xa interacts with users using artificial intelligence and voice recognition. It can be used to perform daily tasks such as making to-do lists, reporting the news and weather, and interacting with other smart devices in the home. In , the Ama-zon Ale-xa app was downloaded some 2800 times per day from the Goo-gle Play store (AppBrain website). Assume that the number of downloads per day of the Ama-zon Ale-xa app is normally distributed with a mean of 2800 and standard deviation of 860. What is the probability there are 2000 or fewer downloads of Ama-zon Ale-xa in a day (to 4 decimals)? What is the probability there are between 1500 and 2500 downloads of Ama-zon Ale-xa in a day (to 4 decimals)? What is the probability there are more than 3000 downloads of Ama-zon Ale-xa in a day (to 4 decimals)? Assume that Goo-gle has designed its servers so there is probability 0.01 that the number of Ama-zon Ale-xa app downloads in a day exceeds the servers' capacity and more servers have to be brought online. How many Ama-zon Ale-xa app downloads per day are Goo-gle's servers designed to handle (to the nearest whole number)?
Gary has 23 quarters in his bank he saves 4 more quarters each week what will the total value of Gary’s bank be in four weeks??
Answers
Answer:
39 quarters, or 9.75 dollars.
Step-by-step explanation:
We can turn this into an algebra equation.
23 + 4x If x = 4
x stands for the number of weeks.
After solving that, we get 39 quarters.
To turn it into dollars, we divide it by 4 (because 4 quarters is equal to one dollar) and then we get 9 3/4. 3/4 dollars is just .75 cents, so the answer is 9.75 dollars.
Garry will have 39 quarters or $9.75 in his bank as per linear equation in 4 weeks.
What is a linear equation?A linear equation is an equation where the variable has the highest power of one.
Give, Garry has 23 quarters.
He saves 4 quarters each week for the next weeks.
Therefore, after 4 weeks Garry will save a total of (4 × 4) quarters = 16 quarters.
Hence, after 4 weeks Garry will have in his bank account is (23 + 16) quarters = 39 quarters = $9.75.
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he probability that michael makes a free throw is 0.45. which probability distribution represents the number of free throws made when michael makes two free throw attempts in a row? responses x012 p0.450.24750.3025x 0 1 2 p 0.45 0.2475 0.3025 , x012 p0.30250.24750.45x 0 1 2 p 0.3025 0.2475 0.45 , x012 p0.20250.4950.3025x 0 1 2 p 0.2025 0.495 0.3025 , x012 p0.30250.4950.2025x 0 1 2 p 0.3025 0.495 0.2025 ,
Answers
the probability distribution for Michael consistently making two free throw attempts: P (X = 0, 1, 2) = 0.3025, 0.495, and 0.2025, respectively.
The probability distribution displays the likelihood that various conceivable experiment results will occur.
When there are numerous attempts at an event with varying probabilities of success and failure, such as two attempts at free throws with a success probability of 0.45
Then it is appropriate to utilize the Binomial probability formula, where Prob =NcR.Pr.Q(n-r): N = Number of trials, R = Number of successes, P = Success probability, and Q = Failure probability.
Specifically, P (success = 0) = 0.3025, P (success = 1) = 0.495, and P (success = 2) = 0.2025
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Suppose that the individuals are divided into groups j = 1,..., J each with n, observations respectively, and we only observe the reported group means y, and j. The model becomes ÿj = Bīj + Uj, (2) Derive an expression for the standard error of the OLS estimator for 3 in terms of ij and Tij indicates ; of individual i belonging to group j. (6 marks) σ, where What are the consequences of heteroskedasticity in the errors for the OLS estimator of the param- eters, their usual OLS standard errors reported by statistical packages, and the standard t-test and F-test for these parameters? (4 marks)
Answers
Heteroskedasticity in the errors has an impact on the accuracy of the standard errors estimated using Ordinary Least Squares (OLS) and can affect hypothesis tests. To address this concern, it is advisable to utilize robust standard errors, which provide more reliable inference regarding the parameters of interest.
In the presence of heteroskedasticity, the OLS estimator for the parameters remains unbiased, but the usual OLS standard errors reported by statistical packages become inefficient and biased. This means that the estimated standard errors do not accurately capture the true variability of the parameter estimates. As a result, hypothesis tests based on these standard errors, such as the t-test and F-test, may yield misleading results.
To address heteroskedasticity, robust standard errors can be used, which provide consistent estimates of the standard errors regardless of the heteroskedasticity structure. These robust standard errors account for the heteroskedasticity and produce valid hypothesis tests. They are calculated using methods such as White's heteroskedasticity-consistent estimator or Huber-White sandwich estimator.
In summary, heteroskedasticity in the errors affects the accuracy of the OLS standard errors and subsequent hypothesis tests. To mitigate this issue, robust standard errors should be employed to obtain reliable inference on the parameters of interest.
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2 apples cost 2 dabloons.
How much does 1 apple cost
Answers
Please help me I beg <3
Answers
5/10 = 0.50
3/4 = 0.75
2/8 = 1/4 = 0.25
Find the measure of line segment MK. K 8 I 7 J M 5 2x+1
Answers
Hello!
This is a problem about relating circle theorems to line lengths.
We can first see that both line segment MK and CM are secants within the circle that come from a common point K.
This means that the Intersecting Secant Theorem applies here.
The Intersecting Secant Theorem states that if two secants are formed from a common point outside the circle, the length of each secant multiplied by the length of its corresponding external secant are equivalent.
We can set up the following equation.
\(8(8+x+5)=7(7+2x+1)\)
\(8(x+13)=7(2x+8)\)
\(8x+104=14x+56\)
\(6x=48\)
\(x=8\)
Using this value, we can find the length of line segment MK.
\(MK=x+5+8\)
\(MK=8+5+8\)
\(MK=21\)
Hope this helps!
I need this done by tomorrow
Answers
Answer:
At least be nice about it bruh.
Make p the subjet in 2/r+1/p=1/q
Answers
well, let's notice the denominators, hmmm so we can use as the LCD say the expression of "rpq", so, let's multiply both sides by the LCD of "rpq" to do away with the denominators.
\(\cfrac{2}{r}+\cfrac{1}{p}=\cfrac{1}{q}\implies \stackrel{\textit{multiplying both sides by } ~~ \stackrel{LCD}{rpq}}{rpq\left( \cfrac{2}{r}+\cfrac{1}{p} \right)=rpq\left( \cfrac{1}{q} \right)}\implies 2pq+1rq=1rp \\\\\\ 2pq+rq=rp\implies rq=rp-2pq\implies rq=\stackrel{\textit{common factoring}}{p(r-2q)}\implies \cfrac{rq}{r-2q}=p\)
The owner offers a payment plan where the total cost of the mountain
bike is paid in 3 equal monthly payments. Determine the amount of each monthly
payment. Show your work or explain your answer. *
Answers
Answer:
Step-by-step explanation:
Write tan 41π/36 in terms of the tangent of a positive acute angle.
Answers
tan(41π/36) can be written in terms of the tangent of a positive acute angle as (tan((1/9)π) + tan((37/36)π)) / (1 - tan((1/9)π)tan((37/36)π))
To express tan(41π/36) in terms of the tangent of a positive acute angle, we need to find an angle within the range of 0 to π/2 that has the same tangent value.
First, let's simplify 41π/36 to its equivalent angle within one full revolution (2π):
41π/36 = 40π/36 + π/36 = (10/9)π + (1/36)π
Now, we can rewrite the angle as:
tan(41π/36) = tan((10/9)π + (1/36)π)
Next, we'll use the tangent addition formula, which states that:
tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
In this case, A = (10/9)π and B = (1/36)π.
tan(41π/36) = tan((10/9)π + (1/36)π) = (tan((10/9)π) + tan((1/36)π)) / (1 - tan((10/9)π)tan((1/36)π))
Now, we need to find the tangent values of (10/9)π and (1/36)π. Since tangent has a periodicity of π, we can subtract or add multiples of π to get equivalent angles within the range of 0 to π/2.
For (10/9)π, we can subtract π to get an equivalent angle within the range:
(10/9)π - π = (1/9)π
Similarly, for (1/36)π, we can add π to get an equivalent angle:
(1/36)π + π = (37/36)π
Now, we can rewrite the expression as:
tan(41π/36) = (tan((1/9)π) + tan((37/36)π)) / (1 - tan((1/9)π)tan((37/36)π))
Since we are looking for an angle within the range of 0 to π/2, we can further simplify the expression as:
tan(41π/36) = (tan((1/9)π) + tan((37/36)π)) / (1 - tan((1/9)π)tan((37/36)π))
Therefore, tan(41π/36) can be written in terms of the tangent of a positive acute angle as the expression given above.
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Найдите значение выражения x − 3(х + 8) при x = −10.
Answers
Answer:
-4
Step-by-step explanation:
Find the range of each set of data. Which city's data has a greater range?
80, 84, 83, 77, 71, 94, 89, 94, 88, 77, 86, 90, 87
48, 35, 82, 87, 64, 35, 89, 34, 56, 44, 75, 56, 72
Answers
Thus, the second set of data has a greater range than the first set, with a range of 55 compared to a range of 23.
The range of a set of data is the difference between the highest and lowest values in that set. To find the range of the first set of data, we need to first find the highest and lowest values. The lowest value in the set is 71, and the highest value is 94.
Therefore, the range is:
Range = Highest value - Lowest value
Range = 94 - 71
Range = 23
To find the range of the second set of data, we need to again find the highest and lowest values. The lowest value in the set is 34, and the highest value is 89.
Therefore, the range is:
Range = Highest value - Lowest value
Range = 89 - 34
Range = 55
Comparing the two ranges, we can see that the second set of data has a greater range than the first set. The range of the second set is 55, while the range of the first set is 23. Therefore, we can conclude that the second city's data has a greater range.
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