Answers
8 = d.
So 8 x pi hot 3.14 in this case) = circumference = 8pi or 25.12
Because this is to find the semicircle it is 8pi/2 or 25.12/2.
=
4pi or 12.56.
This is because the arc of the semicircle is half of the full circle.
Answer: 4pi or 12.56
Related Questions
Nine less than four times a number equals fithteen
Answers
Answer:6
Step-by-step explanation:
let's write this as equation,
4x-9=15
4x=15+9=24
x=24/4=6
a. Is this tunction increasing of decreasing for the yearn2000−2015? A) Incroasing B) Decreasing
Answers
The correct answer is option A) Increasing.
The given question is regarding the function and its behavior. To determine whether a given function is increasing or decreasing, we should first find its derivative. Then, if the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing.A function is given as: yearn = f (2000, 2015)Here, the independent variable is "year," and the dependent variable is "n." Thus, to find out whether this function is increasing or decreasing, we need to find the derivative of this function:$$\frac{dy}{dx}=\frac{dy}{dn}*\frac{dn}{dx}$$$$\frac{dy}{dx}=0.028$$Since the derivative is positive, it means the function is increasing. Therefore, the correct answer is option A) Increasing.
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A student writes down the following piece of work which contains an error in their reasoning. The equation is: y 2
−23y=0 Add 23y to each side: y 2=23y Divide each side By y : y=23 (i) Substitute y=23 into the left-hand side of the equation y 2−23y=0, and explain why this shows that y=23 is indeed a solution of the equation. (ii) Write out a complete solution of the equationy 2−23y=0. (iii) Explain, as if directly to the student, why their solution is inadequate. Solve the following equation, checking that your solution is correct: 6−x16 = x+356
Answers
The following are the answers:
(e) All the parts are discussed in step 2.
(f) The solution is x= 4 and the verification is shown below
How to solveGiven the equation y^2 - 23y = 0
We would follow the steps
Given the equation [tex]\frac{16}{6-x} = \frac{56}{x + y}[/tex]
We need to solve this and verify the solution
(e) (i) Substitute y = 23 on the left-hand side of the given equation.
23^2 -23(23) = 0
0 = 0
Since the left-hand side is equal to the right-hand side, hence y= 23 is the solution of the given equation, but it is not the only solution.
(ii) Complete solution of the given equation:
y^2 -23y = 0
y(y-23) =0
y= 0
and y -23 = 0
y = 0, 23
Hence the complete solution of the given equation is y= 0, 23
EXPLANATION
If a.b = 0, then a= 0, b=0
.
(iii) The solution is inadequate because we can cancel the term only if it non zero.
Also, the given equation has power 2, but the student is getting only a single value, which is incorrect.
Solve the given equation as follows:
[tex]\frac{16}{6-x} = \frac{56}{x + 3}[/tex]
16 (x +3)= 56(6-x)
= 2x + 6= 42- 7x
9x = 36
x= 4
Hence the required solution is x= 4
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PLEASE HELP!!
Write an inequality for the following problem. Use W for your variable.
2 times a number increased by 23 is at most 70.
Answers
Answer:
Step-by-step explanation:
2W+23 is at most 70. So, 2W+23 must be less than or equal to 70.
2W+23≤70
Solve
2W≤47
W≤47/2 or W≤23.5
15 POINTS. How to do this one please teach by step by step
Answers
Given the ASA postulate, it is seen that ABC and DEF are congruent
What is the ASA congruence theorem?
The ASA congruence theorem is a postulate in Euclidean geometry that states that if two angles and the side between them in one triangle are congruent to the corresponding angles and sides in another triangle, then the two triangles are congruent.
More formally, the statement of the ASA congruence theorem can be summarized as follows:
If in two triangles, two pairs of corresponding angles are congruent, and the included sides are also congruent, then the triangles are congruent.
angles ABC and DEF are congruent because they have the same size and shape.
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Can you solve for the other x please
Answers
The solution to the inequality is: x >= 55 or x <= -32.
what is inequality?
An inequality is a mathematical statement that compares two values or expressions and indicates that they are not equal. In other words, an inequality shows the relationship between two values or expressions that are not the same.
To solve the inequality, we need to isolate x on one side of the inequality symbol in each of the two inequalities.
For the first inequality:
3 - 2/5 * x <= -19
Subtracting 3 from both sides, we get:
-2/5 * x <= -22
Dividing both sides by -2/5 (which is the same as multiplying both sides by -5/2), we get:
x >= 55
For the second inequality:
-3/4 * x >= 24
Dividing both sides by -3/4 (which is the same as multiplying both sides by -4/3), we get:
x <= -32
Therefore, the solution to the inequality is:
x >= 55 or x <= -32.
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Which equation represents this transformation?
Answers
This is because the unit is going 3 units right on the x-axis making it a positive (+3) and then subtracting 1 from the y-axis
26. Complete the following proof. Given: \( \angle Q P S \cong \angle T P R \) Prove: \( \angle Q P R \cong \angle T P S \)
Answers
The proof of the congruence of the angles ∠QPS and ∠TPR is shown below
Completing the proofs of the anglesGiven that
∠QPS ≅ ∠TPR
We have the proof of the congruence of the angles ∠TPS and ∠QPR using the following two column style of proofs
Statements Reasons
a ∠QPS ≅ ∠TPR Given
b ∠QPS ≅ ∠TPR Given
c ∠QPS ≅ ∠QPR + ∠RPS Addition property
∠TPR ≅ ∠TPS + ∠RPS
d ∠TPR ≅ ∠QPR + ∠RPS Substitution
e ∠QPR + ∠RPS ≅ ∠TPS + ∠RPS Substitution
f ∠QPS ≅ ∠TPS Subtract property
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Complete question
Complete the following proof. Given: ∠QPS ≅ ∠TPR
Prove: ∠QPS ≅ ∠TPS
Statements Reasons
a ______________ ______________
b ∠QPS ≅ ∠TPR ______________
c ∠QPS ≅ ∠QPR + ∠RPS ______________
∠TPR ≅ ∠TPS + ∠RPS
d ______________ Substitution
e ______________ ______________
f ______________ ______________
Find the Z-scores for which 50% of the distribution's area lies between -z and z.
Click to view page 1 of the table Click to view page 2 of the table.
The z-scores are
(Use a comma to separate answers as needed Round to two decimal places as needed. )
Answers
The z-scores for which 50% of the distribution's area lies between -z and z are: z = -0.675 and z = 0.675.
How to obtain probabilities using the normal distribution?The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.The normal distribution is symmetric, meaning that the middle 50% of the scores is given between these following percentiles:
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1. There are three buses A, B and C. the total
number of seats in the three buses is 250. 28% of
the 250 seat are in bus A.
number of seats in bus B:number of seats in bus
C = 3:2
38 of the seats in bus B are empty. 26 in bus C
are empty.
Jamie says the percentage of the seats in bus B
that are empty is greater than the percentage of
the seats in bus C that are empty. Is Jamie
correct.
Answers
Thus, Jamie is wrong. In contrast to Bus C, Bus B has less unoccupied seats as a percentage of total seats.
How do the percentages translate?% is a relative number that is used to represent hundredths of the any quantity. Since one percent (symbolized as 1%) is equal to one hundredth of something, 100 percent stands for everything, and 200 percent refers to twice the amount specified. percentage.
Find the complete seating capacity in Bus A first:
Bus A has 28% of the available seats, making its seat count 0.28 x 250 = 70.
Secondly, we can construct an equation system using the number of seats in Bus B compared to Bus C:
The number of seats aboard Bus B will be x.
In that case, Bus C's seating capacity is (2/3)x.
Using the information that there are 250 seats overall, we can find x:
70 (from Bus A) + x (from Bus B) + (2/3)x (from Bus C) = 250
When we simplify this equation, we obtain: (5/3)x = 180
When we solve for x, we get x = 108.
Bus C has (2/3)x = 72 seats, but Bus B has 108 seats.
We can now determine the proportion of vacant seats for each bus:
Out of a total of 108 seats, 38 seats on Bus B are vacant, making the percentage of vacant seats in Bus B (38/108) x 100% = 35.19%.
Out of a total of 72 seats, there are 26 empty seats on Bus C, making the percentage for empty seats there (26/72) x 100% = 36.11%.
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Answer:
what paper is this for this question
Every morning, Matthew ills his dog’s water dish with 16 oz of water. If his dog finishes his water every day, how many ounces will his dog drink in a week?
How many cups is this?
How many pints is this?
How many quarts is this equal to?
How many quarts is this?
Answers
The dog drinks 112 ounces, or 14 cups, or 7 pints, or 3.5 quarts of water per week.
How to find intake of the dog?The dog drinks 16 oz of water every day, so in a week (7 days), the dog will drink:
16 oz/day × 7 days/week = 112 oz/week
To convert ounces to cups, we divide by 8 (since there are 8 fluid ounces in a cup):
112 oz/week ÷ 8 oz/cup = 14 cups/week
To convert ounces to pints, we divide by 16 (since there are 16 fluid ounces in a pint):
112 oz/week ÷ 16 oz/pint = 7 pints/week
To convert ounces to quarts, we divide by 32 (since there are 32 fluid ounces in a quart):
112 oz/week ÷ 32 oz/quart = 3.5 quarts/week
Therefore, the dog drinks 112 ounces, or 14 cups, or 7 pints, or 3.5 quarts of water per week.
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Raju & akash is given to solve a mathematical problem. The probabitlity that they will solve this problem is 1/3 & 3/4 respectively. Then, find the probability that both Raju & AKAsh will solve any random problem given to them after sufficient time
Answers
The probability that both Raju and Akash will solve any random problem given to them after sufficient time is 1/4.
The probability that Raju will solve a random problem given to him is 1/3 and the probability that Akash will solve the same problem is 3/4. We can use the multiplication rule of probability to find the probability that both of them will solve any random problem given to them after sufficient time.
According to the multiplication rule of probability, the probability of two independent events A and B occurring together is the product of their individual probabilities:
P(A and B) = P(A) × P(B)
In this case, the event "Raju solves a problem" is independent of the event "Akash solves a problem". Therefore, the probability that both Raju and Akash will solve a random problem given to them after sufficient time is:
P(Raju and Akash) = P(Raju) × P(Akash)
= 1/3 × 3/4
= 1/4
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Alisa is choosing new tile for the floor in her dining room which is in the shape of a square with side lengths x feet. The tile costs $3. 39 per square foot
Answers
Th function is f(x) = 3.39x²
The cost of the flooring for a dining room is $410.19.
The cost of the flooring for a dining room is $6,072.31.
What is a function?
A function has an input and an output.
A function can be one-to-one or onto one.
It simply indicated the relationships between the input and the output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
Area of a square = side²
Square with side lengths = x feet.
The tile costs = $3.39 per square foot.
a.
The function f for the cost of the flooring.
f(x) = 3.39x²
b.
x = 11 feet
f(11) = 3.39 x 11²
f(11) = 3.39 x 121
f(11) = $410.19
c.
x = 23 feet
f(23) = 3.39 x 23²
f(23) = 3.39 x 529
f(23) = $6,072.31
Therefore,
f(x) = 3.39x²
The cost of the flooring for a dining room is $410.19.
The cost of the flooring for a dining room is $6,072.31.
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Complete Question
Alisa is choosing new tile for the floor in her dining room which is in the shape of a square with side lengths x feet. The tile costs $3.39 per square foot.
a. Write the function f for the cost of the flooring.
b. Determine the cost of the flooring if she decides on a dining room with side lengths of 11 ft.
c. Determine the cost of the flooring if she decides on a dining room with side lengths of 23 ft
Which deal is better. 500ml of lemonade for 94p or 750ml of lemonade for £1.44
Answers
Answer:
To compare the deals, we need to find the price per milliliter of lemonade for each deal.
For the first deal, the price per milliliter is:
94p / 500ml = 0.188p/ml
For the second deal, the price per milliliter is:
£1.44 / 750ml = 0.192p/ml
Therefore, the first deal is better as it has a lower price per milliliter of lemonade.
L: 40 in
The figure will be dilated by a D
scale factor of 3.5. Find the
new measure of the base.
9 in
10 in
9 in
Answers
The new measure of the base is 35 inches
How to determine the new measure of the baseFrom the question, we have the following parameters that can be used in our computation:
Base = 10 inches
Scale factor of dilation = 3.5
Using the above as a guide, we have the following:
Image of the point = Scale factor of dilation * Base
Substitute the known values in the above equation, so, we have the following representation
New measure of base = 3.5 * 10
Evaluate
New measure of base = 35
Hence, the new base is 35 inches
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Fill in the blank question.
The table shows the distance a runner has traveled y, in miles, x minutes after the start of a race. What is the runner's average rate of change, in miles per minute, from 60 to 120 minutes?
Time (min) Distance (miles)
0 0
30 3.48
60 6.42
90 9.18
120 12.0
Answers
According to the speed formula, the runner's average rate of change from 60 to 120 minutes is approximately: 5.58 / 60 = 0.093 miles per minute
What is speed?
Speed is defined as the distance travelled by an object in a given amount of time. Speed is a scalar quantity, meaning that it has magnitude but no direction.
Mathematically, speed is calculated as follows:
speed = distance / time
where "distance" is the distance travelled by the object, and "time" is the time it takes for the object to travel that distance.
The average rate of change of a function over an interval is the slope of the secant line between the endpoints of the interval.
In this case, we want to find the average rate of change of the distance function from 60 to 120 minutes. The distance function is given by the table as:
Time (min) Distance (miles)
0 0
30 3.48
60 6.42
90 9.18
120 12.0
To find the average rate of change from 60 to 120 minutes, we need to calculate the slope of the secant line passing through the points (60, 6.42) and (120, 12.0).
The slope of the secant line is given by:
slope = (change in y) / (change in x) = (12.0 - 6.42) / (120 - 60) = 5.58 / 60
Therefore, the runner's average rate of change from 60 to 120 minutes is approximately: 5.58 / 60 = 0.093 miles per minute
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III. Three fair tetrahedra (same as the one in part II except (magenta; green; red)) are tossed. Find the prob's of: a. All the same Greek letter. b. No γ 's nor δ 's. c. Exactly two β 's. d. All different Greek letters.
Answers
Answer:
d all differet grreek letters
(1 point) Assume that the monthly wondwide average number of airplaine crashes of commercial ailines is \( 2.2 \). What is the probability that there wili be (a) at most 3 such accidents in the next m
Answers
Answer: 2.2
Step-by-step explanation:
A softball coach has ordered softballs for two different leagues. The Junior League uses an 11-inch softball priced at $2.50 each. The Senior League uses a 12-inch softball priced at $3.50 each. The softball coach ordered a total of 120 softballs for $350.
If j is the number of softballs ordered for the Junior League and s is the number of softballs ordered for the Senior League, select the system of linear equations that represents the situation.
1. j + s = 350
2.5j + 3.5s = 120
2. j + s = 120
3.5j + 2.5s =350
3. j + s = 350
3.5j + 2.5s = 120
4. j + s = 120
2.5j + 3.5s = 350
Answers
The system of linear equations that represents the situation is:
(option 4) j + s = 120
2.5j + 3.5s = 350
How to know the system of linear equations that represents the situation?Let j be the number of 11-inch softballs ordered for the Junior League, and s be the number of 12-inch softballs ordered for the Senior League.
The total number of softballs ordered is 120, so we have:
j + s = 120
The cost of each 11-inch softball is $2.50, and the cost of each 12-inch softball is $3.50. The total cost of the order is $350, so we have:
2.5j + 3.5s = 350
Therefore, the system of linear equations that represents the situation is:
j + s = 120
2.5j + 3.5s = 350
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Find the indicated real nth roots of a n=3, a=-125
Answers
The cube root of -1 is -1, and the cube root ∛(-125) is -5.
To find the real nth roots of a number a, we can use the formula:
[tex]√(a) = a^(1/n)[/tex]
For the case where n=3 and a=-125, we have:
[tex]√(-125) = (-125)^(1/3)[/tex]
We can simplify this using the fact that
[tex](-a)^(1/n) = -(a^(1/n)):[/tex]
√(-125) = - (√125)
Now we need to find the cube root of 125. We can factor 125 as 555, so:
√(-125) = - (√125) = - (√(555)) = - (5√5)
Therefore, the real cube root of -125 is -5√5.
The cube root of -125 is the real number x that satisfies the equation
[tex] {x}^{3} [/tex]
= -125.
We can rewrite -125 as -1*5^3, which means we can write the cube root of -125 as ∛(-1)*∛(5^3).
The cube root of -1 is -1, and the cube root of 5^3 is 5, so ∛(-125) = -1*5 = -5.
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100 points please help!!
Answers
Answer: (5x-1)(x+4)
Step-by-step explanation:what she said
What is the measure of angle H? Explain what is the measure of angle i ? explain
Answers
The measure of angle H is given as 67.5 degrees
The measure for angle I is 112.5 degrees
How to solve for angle HThe question tells us that ∠F must be 67.5°
from the sign, the angles H and the angle F are vertical angles.
Hence the value of ∠F = ∠H
∠H = 67.5°
what is the measure of angle i ?From the transversal, we can see that the angle E and angle I are corresponding angles
to get angle E
= 67.5° + ∠E = 180 (angles on a straight line)
∠E = 180 - 67.5
∠E = 112.5 degrees
Since ∠E and ∠I are corresponding angles, ∠I = 112.5 degrees
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The stop sign is a regular octagon, so the measure of ∠F must be 67.5°. What is the measure of angle H? Explain what is the measure of angle i ? explain
Describe the end behavior of each function.
1. f(x) = -x³ + 2x³ -x+4
2. f(x)= x² -x² -x -1
Asking for help please, Thank you
Answers
"1. The end behavior of the function f(x) = -x³ + 2x³ -x+4 can be determined by looking at the highest degree term, which is 2x³. Since this term is positive and has an even degree, the end behavior of the function will be the same on both ends. As x approaches negative infinity, the function will approach negative infinity, and as x approaches positive infinity, the function will approach positive infinity. Therefore, the graph of the function will have two arms pointing upwards.
2. The function f(x) = x² -x² -x -1 can also be analyzed by looking at the highest degree term, which is x². However, since the first two terms cancel each other out, we can simplify the function to f(x) = -x -1. As x approaches negative infinity, the function will approach negative infinity, and as x approaches positive infinity, the function will approach negative infinity as well. Therefore, the graph of the function will have a single arm pointing downwards." (ChatGPT, 2023)
Lily is practicing multiplying complex numbers using the complex number (2+i).
To determine the value of (2+i)²,Lily performs the following operations:
Step 1: (2+i)²= 4+i²
Step 2: 4+i² = 4+ (−1)
Step 3: 4+(-1)=3
Lily made an error.
Explain Lily's error and correct the step which contains the error.
Bonus:
Lily is continuing to explore different ways in which complex numbers can be multiplied so the
answer is not a complex number. Lily multiplies (2+i) and (a+bi), where a and b are real
numbers, and finds that her answer is not a complex number.
A. Write an equation that expresses the relationship between a and b.
Answers
Lily's error is in the first step, (2+i)^2 ≠ 4 + i^2.
(2+i)^2 = (2+i)(2+i) and you need to FOIL.
(2+i)^2 = (2+i)(2+i)
= 4 + 2i + 2i + i^2
= 4 + 4i + (-1)
= 3 + 4i
Bonus:
If you want to multipy (2+i) by (a+bi) and not end up with a complex number, you'd first FOIL
(2+i)(a+bi) = 2a + 2bi + ai + bi^2
We know i^2 = -1, so this becomes
= 2a + 2bi + ai + b(-1)
= 2a + 2bi + ai - b
= 2a - b + 2bi + ai
Now, for this not to be complex, we need the imaginary pieces to cancel each other out. In other words 2bi+ai=0. For that to happen, 2b + a = 0, or
2b = -a or a = -2b
So it would seem that if we pick any b-value and make a = -2b, then we'll end up with a non-complex number.
Let's try b=5, making a = -10
(2+i)(-10+5i) = -20 + 10i - 10i + 5i^2
The 10i's cancel and 5i^2 = -5, so we're left just with -25.
When two dice are rolled, the total is between 2 and 12 inclusive. A student simulates the rolling of two dice by randomly generating numbers between 2 and 12. Does this simulation behave in a way that is similar to actual dice? Why or why not?
Answers
The simulatiοn dοes't behave in way that is similar tο actual dice because οf variatiοn οf prοbability.
What is prοbability?Prοbability is a way οf calculating hοw likely sοmething is tο happen. It is difficult tο prοvide a cοmplete predictiοn fοr many events. Using it, we can οnly fοrecast the prοbability, οr likelihοοd, οf an event οccurring. The prοbability might be between 0 and 1, where 0 denοtes an impοssibility and 1 denοtes a certainty.
a)The pοssible cοmbinatiοns frοm the twο dices tο get a tοtal οf seven are
=> (1,6) (2,5) (3,4) (4,3) (5,2) (6,1)
i.e. 6 cases and the tοtal nο. οf cοmbinatiοns
=>6 * 6 = 36.
Hence the nο. οf cases favοurable fοr us οut οf all the cοmbinatiοns is 6.
Hence the prοbability is 6/36 = 1/6.
b)To get a 7 out from a set of 2,12 inclusive is 1/11.
Because it is a choice of 1 no. from the 11 numbers available.
c). NO.
This simulatiοn dοesn't behave like the actual dice because in the secοnd case we are chοοsing οne number frοm the 11 available οptiοns whereas in the first case it is the prοbability οf chοοsing the 6 right cοmbinatiοns frοm the 36 cοmbinatiοns available.
i.e . Chοοsing 6 frοm 36 . hence the prοbability is 1/6.
whereas the chοice οf 1 frοm 11 . i.e prοbability is 1/11.
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A quadrilateral has two angles that measure 216° and 102°. The other two angles are in a ratio of 10:11. What are the measures of those two angles?
Answers
The measures of the missing angles are 20° and 22°.
The sum of the angles of a quadrilateral is 360 degrees. We can use this fact to find the measure of the missing angles.
Let x and y be the measures of the missing angles, such that x:y = 10:11. Then we have:
216° + 102° + x + y = 360°
Simplifying this equation, we get:
x + y = 42°
We also know that x:y = 10:11, so we can write:
x = 10k
y = 11k
where k is a constant. Substituting these expressions into the equation x + y = 42°, we get:
10k + 11k = 42
21k = 42
k = 2
Therefore, x = 10k = 20° and y = 11k = 22°.
So, the measures of the missing angles are 20° and 22°.
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for a normal distribution with mean 50 and standard deviation 15
find the Q1 and Q3
Answers
Q1 = 7.75th item
Q3 = 23.25th item
Given,For a normal distribution with mean 50 and standard deviation 15,To find Q1 and Q3Formula for Q1 and Q3:Q1 = (n + 1)/4Q3 = 3(n + 1)/4Here, n = total number of observationsn = 2 × 15 = 30Thus, n + 1 = 31We know that for a normal distribution the Z values for Q1 and Q3 are -0.67 and 0.67 respectively.Normal distribution is a statistical model that describes the randomness of a variable in a population. It is symmetric around the mean. The mean is an essential property of a normal distribution. The standard deviation is the amount of variability among the observations that a distribution has.Mean = 50Standard deviation = 15Q1 = (n + 1)/4 = (30 + 1)/4 = 7.75th itemQ3 = 3(n + 1)/4 = 3 × (30 + 1)/4 = 23.25th itemThe 7.75th item and 23.25th item are calculated with the help of the formula.
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When the outlier(s) are removed, how does the mean change?
A) The mean decreases by 1.9.
B) The mean increases by 2.4.
C) The mean increases by 1.9.
D) There are no outliers.
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Answer:
B) The mean increases by 2.4 because the mean is 1149 and there 6 outliers.
can someone explain how to solve this question with steps
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First we need to find the inverse function. To do this, I'd put f(x) into vertex form:
[tex]\begin{aligned}f(x) &= (x^2-4x)-6\\[0.5em] &= (x^2-4x+4)-6-4\\[0.5em] &= (x-2)^2-10\\[0.5em]\end{aligned}[/tex]
Now, we can try to invert the function, keeping in mind this is only used when x≥2.
[tex]\begin{aligned}y&= (x-2)^2-10\\[0.5em]y+10&= (x-2)^2\\[0.5em]\sqrt{y+10}&= x-2\\[0.5em]\sqrt{y+10}+2&= x\\[0.5em]\end{aligned}[/tex]
(The fact that x ≥ 2 allowed us only keep the positive square root on the third line.)
So our inverse function is [tex]f^{-1}(y)=\sqrt{y+10}+2[/tex].
Now, let's find the derivative of this function:
[tex]\begin{aligned}\dfrac{df^{-1}}{dy}&= \dfrac{1}{2\sqrt{y+10}}\cdot\dfrac{d}{dy}[y+10]\\[0.5em] &= \dfrac{1}{2\sqrt{y+10}}\end{aligned}\\[/tex]
(The chain rule was used, but the derivative of the inside function equals 1.)
So [tex](f^{-1})'(-6) = \dfrac{1}{2\sqrt{-6+10}} = \dfrac{1}{4}[/tex]
Now having done all of that, it did cross my mind that since [tex]f[/tex] and [tex]f^{-1}[/tex] are simply reflections over the line y=x, if we had actually just found f'(4) and then used the reciprocal slope, we'd also get the same answer more quickly.
f'(x) = 2x - 4
when y = -6, then x = 4 (by solving f(x) = -6).
f'(4) = 4, so reflecting that slope of 4/1 over the line y=x, we'd get a slope of 1/4.
PLEASE SHOW WORK!!!!!!!!!
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Answer:
The answer is D