Answers
ANSWER
k = 1 or 21
STEP-BY-STEP EXPLANATION:
According to the question, we were given the below trigonometric function
[tex]\sec ^2x\text{ - 22tanx + 20 = 0}[/tex]Recall that, we have trigonometric identity which is written below
[tex]\sin ^2\theta+cos^2\theta\text{ = 1}[/tex][tex]\text{Divide through by }\cos ^2\theta[/tex][tex]\begin{gathered} \frac{\sin^2\theta}{\cos^2\theta}\text{ + }\frac{cos^2\theta}{\cos^2\theta}\text{ =}\frac{1}{\cos ^2\theta} \\ \tan ^2\theta+1=sec^2\theta \\ \text{Let x = }\theta \\ \tan ^2x+1=sec^2x \end{gathered}[/tex]The next thing is to rewrite the equation
[tex]\begin{gathered} \text{ since sec}^2x=tan^2x\text{ + 1} \\ \text{Hence,} \\ \tan ^2x\text{ + 1 - 22tanx + 20 = 0} \\ \text{Let k = tanx} \\ k^2\text{ + 1 -22k + 20 = 0} \\ \text{Collect the like terms} \\ k^2\text{ - 22k + 21 = 0} \end{gathered}[/tex]The next thing is to find the value of P by factorizing the above equation.
Recall that, the standard form of the quadratic function is given as
[tex]ax^2\text{ + bx + c = 0}[/tex]Let
a = 1
b = -22
c = 21
The next thing is to find the value of ac
[tex]\begin{gathered} ac\text{ = 1 }\cdot\text{ 2}1 \\ ac\text{ = 2}1 \end{gathered}[/tex][tex]\begin{gathered} k^2\text{ - k -21k + 21 =0} \\ k(k\text{ -1) -21(}k\text{- 1) = 0} \\ (k\text{ -1) (k -21) = 0} \\ k\text{ -1 = 0 or =k - 21 = 0} \\ k\text{ = 1 or k = 22} \end{gathered}[/tex]Hence, the value of k is 1 or 21
Related Questions
In ∆MNO , o = 790cm. < O=50° and
Answers
Using the law of sines:
[tex]\begin{gathered} \frac{o}{\sin(O)}=\frac{m}{\sin (M)} \\ \frac{790}{\sin(50)}=\frac{m}{\sin (25)} \\ m=\frac{790\cdot\sin (25)}{\sin (50)} \\ m=435.834278 \end{gathered}[/tex][tex]\begin{gathered} \frac{o}{\sin(O)}=\frac{n}{\sin (N)} \\ \frac{790}{\sin(50)}=\frac{n}{\sin (105)} \\ n=\frac{790\cdot\sin (105)}{\sin (50)} \\ n=515.6358793 \end{gathered}[/tex]Using the heron formula:
[tex]\begin{gathered} s=\frac{790+435.834278+515.6358793}{2} \\ s=870.7350787 \\ so\colon \\ A=\sqrt[]{870.7350787(870.7350787-790)(870.7350787-435.834278)(870.7350787-515.6358793)} \\ \end{gathered}[/tex][tex]\begin{gathered} A=104194.335 \\ A=104194cm^2 \end{gathered}[/tex]The patient recovery time from a particular surgical procedure is normally distributed with a mean of 3 days and a standard deviation of 1.7 days. Let X be the recovery time for a randomly selected patient. Round all answers to 4 decimal places where possible.a. What is the distribution of X? X ~ N(,)b. What is the median recovery time? daysc. What is the Z-score for a patient that took 4.1 days to recover? d. What is the probability of spending more than 2.4 days in recovery? e. What is the probability of spending between 2.7 and 3.4 days in recovery? f. The 80th percentile for recovery times is days.
Answers
Given
mean = 3 days
standard deviation = 1.7 days
Find
a. What is the distribution of X?
b. What is the median recovery time?
c. What is the Z-score for a patient that took 4.1 days to recover?
d. What is the probability of spending more than 2.4 days in recovery?
e. What is the probability of spending between 2.7 and 3.4 days in recovery?
f. The 80th percentile for recovery times
Explanation
a) Distribution of X is given by X ~ N( 3 , 1.7)
b) for the normal distibution ,the median is the same as the mean .
so , the median recovery time is 3 days
c) z - score for the patient that took 4.1 days to recover is
[tex]\begin{gathered} z=\frac{X-\mu}{\sigma} \\ \\ z=\frac{4.1-3}{1.7} \\ \\ z=0.64705882352\approx0.6471 \end{gathered}[/tex]d) probability of spending more than 2.4 days in recovery
[tex]\begin{gathered} P(X>2.4)=P(\frac{X-\mu}{\sigma}>\frac{2.4-3}{1.7}) \\ \\ P(X>2.4)=P(Z>-0.3529) \\ P(X>2.4)=P(Z<0.3529) \\ \\ P(X>2.4)=0.6379 \end{gathered}[/tex]e) probability of spending between 2.7 and 3.4 days in recovery
[tex]\begin{gathered} P(2.7f) 80th percentile for recovery times = [tex]\begin{gathered} P(XFinal AnswerHence , the above are the required answers.
The sum of two integers is 463, and the larger number is 31 more than 5 times the smaller number. Findthe two integers.
Answers
SOLUTION
Let the smaller number be x
Let the larger number be y
Since the larger number is 31 more than 5 times the smaller number, it folllows:
[tex]y=5x+31[/tex]The sum of the two integers is 463, it follows
[tex]x+y=463[/tex]Substitute y=5x+31 into the equation
[tex]x+5x+31=463[/tex]Solve for x
[tex]\begin{gathered} 6x=463-31 \\ 6x=432 \\ x=\frac{432}{6} \\ x=72 \end{gathered}[/tex]Substitute x=72 into y=5x+31
[tex]\begin{gathered} y=5\left(72\right)+31 \\ y=360+31 \\ y=391 \end{gathered}[/tex]Therefore the two integers are 72 and 391
evaluate the expression:[tex] \frac{8}{5 - 1} \times (3 + 6) \times 3[/tex]A)102B)-12C)62D)54
Answers
To evaluate the expression:
[tex]\frac{8}{(5-1)}\cdot(3+6)\cdot3[/tex]We can use PEMDAS. We can start from the parenthesis, then, since we have divisions and multiplications, and they have the same precedence, we can start doing the evaluation from the left to the right. Then, we have:
[tex]\frac{8}{(4)}\cdot(9)\cdot3=2\cdot9\cdot3=54[/tex]The answer is 54 (option D).
Find the unit rate.
576 passengers in 144 cars =
Save answer
passengers per car
Answers
The unit rate is 4 passengers per car
We need to find the unit rate.
576 passengers in 144 cars
rate = 576/144
rate = 4 passengers per car
Therefore, the unit rate is 4 passengers per car
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Consider the equation. Determine whether the graph of the equation is wider or narrower than the graph of Y=1/2x^2+1
Answers
It's important to observe that the number that multiples the x is 1/2, that is, it's less than zero.
Jim and Carla are scuba diving. Jim started out 8 feet below the surface. He descended 18 feet, rose 5 feet,and descended 9 more feet. Then he rested. Carla started out at the surface. She descended 16 feet, rose 5feet, and descended another 18 feet. Then she rested. Which person rested at a greater depth? Completethe explanation.
Answers
To solve this question, we need to use integers to express different altitudes. We can sketch each situation as follows:
To find the depth at which each person rested, we need to algebraically sum all these altitudes.
Jim:
-8 -18 + 5 - 9 = -30. He rested at 30 feet below the surface.
Carla:
-16 + 5 - 18 = -29. She rested at 29 feet below the surface.
Therefore, Jim rested at a greater depth (30 feet below the surface).
The radius of a circle is 8 miles. What is the area of a sector bounded by a 180° arc?Give the exact answer in simplest form. ____ square miles.
Answers
Radius r:
r = 8 miles
Area of a circle:
A = π * r²
The area of a 180° arc is the half of the area of the entire circle:
A_arc = (π * r²)/2
Solving:
A_arc = 32π square miles
what percentage of college graduates believe their education was useful for helping them grow personally and intellectually, but not useful for helping them develop specific skills and knowledge for the workplace?Write your answer as a percentage
Answers
We get the following from the given table.
The percentage of college graduates who believes their education was helping them grow personally and intellectually =62 % + 31 %= 93 %
The percentage of college graduates who believes their education was useful for helping them develop specific skills and knowledge for the workplace = 49%+25%= 84 %
The percentage of college graduates who does not believe their education was useful for helping them develop specific skills and knowledge for the workplace=100-84=16%
The percentage of college graduates believes their education was helping them grow personally and intellectually and does not believe useful for helping them develop specific skills and knowledge for the workplace=the least value of 93 and 16 is 16 %.
Hence the required percentage is 16%.
A rectangular patio measures 20 ft by 30 ft. You want to double the patio area by adding the same distance x to the length and width. Write and solve an equation to find a value for x, then find the new dimensions for the patio.
Answers
area = side x side = 20 x 30 =600 ft
we want to double the area, this is 1200 ft, therefore:
[tex]\begin{gathered} \mleft(x+20\mright)\mleft(x+30\mright)=1200 \\ x^2+50x+600=1200 \\ x^2+50x-600=0 \\ \mleft(x+60\mright)\mleft(x-10\mright)=0 \\ x+60=0 \\ x=-60 \end{gathered}[/tex]but, We don't measure in negative numbers, so we disregard this answer. and the other
[tex]\begin{gathered} x-10=0 \\ x=10 \end{gathered}[/tex]therefore, the patio is now 30ft x 40ft
In the diagram, M is the midpoint of BD and AC. Name two triangles that are congruent.BADABMCX ADMAA ABDACBAA CDM ABMADAB ABCD
Answers
From the question,
The two triangles that are congruent are :
Triangle CDM = Triangle ABM --------- Option C
find al the solutions for x.9. 8x2+19 = 54 +3x
Answers
8x^2 + 19 = 54 + 3x^2
Solving for x:
8x^2 - 3x^2 = 54 - 19
5x^2 = 35
x^2 = 35/5 = 7
x^2 = 7
x = sqrt(7) = 2.6458
x = -sqrt(7) = -2.6458
Answer:
It has two solutions:
x = 2.6458 and x = -2.6458
PLEASE HELP!!!!!!!!
Choose the two graphs that preserve congruence.
Answers
The two graphs that reserve congruence are in the option A and option D
What is congruency?Congruent refers to something that is "absolutely equal" in terms of size and shape.
The shapes hold true regardless of how we rotate, flip, or turn them.
Draw two circles with the same radius, for instance, cut them out, and stack them on top of one another. We will see that they will superimpose, or be positioned entirely on top of, one another. This demonstrates the congruence of the two circles.
The images on the graphs in the options A and options D both maintains congruence, The sizes are identical and has a rigid transformation
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the table below shows the minimum volume of water needed to fight a typical fire in rooms of various sizes. Find the rate of change. Explain the meaning of rate of change. Include the units in your answer.
Answers
Since the volume of water depends on the floor area, the floor area is the independent variable while the minimum water volume is the dependent variable.
Let x represents the floor area
Let y represent the miminum volume of water
The rate of change is the ration of the change in y to the change in x
Let the rate of change be represented by dy/dx
[tex]\begin{gathered} \frac{dy}{dx}=\text{ }\frac{y_2-y_1}{x_2-x_1} \\ \end{gathered}[/tex]Considering the first two rows of the table, since the rate of change does not differ despite the rows picked
[tex]\begin{gathered} x_1=25,y_1=39,x_2=50,y_2=78 \\ \frac{dy}{dx}=\text{ }\frac{78-39}{50-25} \\ \frac{dy}{dx}=\frac{39}{25} \\ \frac{dy}{dx}=1.56 \end{gathered}[/tex]Determine the value of k for which the inequality $0.5<-4x+k\le12-k$ has the solution set $\left\{x|1.25\le x<2\right\} Need the answer ASAP.
Answers
The value of k for which the inequality has the given solution set is:
k = 8.5.
How to obtain the value of k?The inequality is presented as follows:
0.5 < -4x + k ≤ 12 - k.
Two inequalities compounded, hence the and operation is applied, which means that the solution set is composed by the elements that respect both conditions.
The solution set of the inequality is:
1.25 ≤ x < 2.
The lower bound of the solution is of 1.25, hence:
-4x + k ≤ 12 - k.
-4x + 2k ≤ 12
4x ≥ 2k - 12
x ≥ 0.5k - 3
Hence:
0.5k - 3 = 1.25
0.5k = 4.25
k = 4.25/0.5
k = 8.5.
The upper bound of the solution is of 2, hence:
-4x + k > 0.5
-4x > 0.5 - k
x < -0.125 + 0.25k
Hence:
-0.125 + 0.25k = 2
k = 2.125/0.25
k = 8.5. -> Which confirms the solution.
Missing InformationThe problem is given by the image shown at the end of the answer.
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The graph above: One to one function Function but not one to one Relation but not a function
Answers
The given graph represents a function, also notice that it is a one to one function because using the vertical line theorem we have that for all vertical lines it only intercepts the graph in just one point.
How many gallons each of 25% alcohol and 10% alcohol should be mixed to obtain 15 gal of 21% alcohol?Gallons ofPure AlcoholGallons ofSolutionХy15Percent(as a decimal)25% = 0.2510% = 0.121% =How many gallons of 25% alcohol should be in the mixture? gal
Answers
Answer
11 gallons of the 25% alcohol is required for the mixture.
4 gallons of the 10% alcohol is required for the mixture.
Explanation
Let the number of gallons of 25% alcohol required be x
Let the number of gallons of 10% alcohol required be y
The total amount of gallons required is 15 gallons. In mathematical terms,
x + y = 15
The alcohol content of the 15 gallons is to be 21%.
21% of 15 gallons = 3.15 gallons
From the first statements,
Let the number of gallons of 25% alcohol required be x
Let the number of gallons of 10% alcohol required be y
25% of x gallons = 0.25x gallons
10% of y gallons = 0.10y gallons
0.25x + 0.10y = 3.15
We can then bring these two equations together to solve simultaneously
x + y = 15
0.25x + 0.10y = 3.15
Solving this simultaneously with the calculator, we obtain
x = 11 gallons
y = 4 gallons
Hope this Helps!!!
here is a net of right triangles and rectangles all measurements are given in centimeters.
Answers
Problem
Solution
For this case we we can find the area on this way:
[tex]A=\frac{5\cdot4}{2}+6\cdot4+6\cdot5+6\cdot3+\frac{4\cdot3}{2}[/tex]And solving we got:
[tex]A=10+24+30+18+6=88[/tex]The area for this case is 88 unit^2
a farmer wants to build a fence in the shape of a parallelogram for his animals. The perimeter of the fence will be 600 feet, and the North/South fences are half of the length of the West/East fences. If fences are sold in 5 foot segments, how many fence segments does the farmer need to buy ?
Answers
Let's use the variable L to represent the length and W to represent the width.
If the perimeter is 600 ft, we have:
[tex]\begin{gathered} P=2L+2W\\ \\ 2L+2W=600\\ \\ L+W=300 \end{gathered}[/tex]The width is half the length, so we have:
[tex]\begin{gathered} W=\frac{L}{2}\rightarrow L=2W\\ \\ 2W+W=300\\ \\ 3W=300\\ \\ W=\frac{300}{3}\\ \\ W=100\text{ ft}\\ \\ L=200\text{ ft} \end{gathered}[/tex]Now, if each fence segment is 5 ft, we number of segments needed is:
[tex]\begin{gathered} \text{ fences for W1: }\frac{100}{5}=20\\ \\ \text{ fences for W2: }\frac{100}{5}=20\\ \\ \text{ fences for L1:}\frac{200}{5}=40\\ \\ \text{ fences for L2:}\frac{200}{5}=40\\ \\ \\ \\ \text{ total:}20+20+40+40=120\text{ fence segments} \end{gathered}[/tex]A model airplane is built at a scale of 1 inch to 32 feet. If the model plane is 8 inches long, the actual length of the airplane is blank feet.
Answers
If the model plane is built at a scale of 1 inch to 32 feet
let x be the actual length of the airplane
1/32 = 8/x
cross-multiply
x = 32 times 8
x = 256 inches
Billy used four colors to divide 4.20 by 4. Which model shows 4.20/4?
Answers
Given that Billy used four colours to divide 4.20 by 4
To Determine: The model that shows this division
Solution:
It can be observed that 1 represents 100 small boxes
Dividing 4.20 by 4 would give
[tex]\frac{4.20}{4}=1.05[/tex]1.05 would be represented by small boxes
[tex]1.05\times100=105[/tex]If we used colours, then each of the colours would have 105 boxes
From the above explanation, the model that shows 4.20/4 is OPTION D
In the diagram, BAE is a semicircle, and mZACE = 28° . Based on your explorations, which of the following statements must be true. Select all that apply.
Answers
Since the arc BAE is a semicircle, it measures 180°, and the angle that inscribes it, that is, angle ∠BCE, has half the measure, so ∠BCE = 90°.
The angle ∠ACE inscribes the arc AE, so the arc AE has double the measure of the angle ∠ACE, so AE = 56°.
Calculating the measure of the arc AB, we have:
[tex]\begin{gathered} AB+AE=BAE \\ AB+56=180 \\ AB=180-56 \\ AB=124\degree \end{gathered}[/tex]So the first option is correct.
For the second option, these two angles inscribes the same arc (arc AE), so they have the same measure of half the measure of the arc. Therefore, they are congruent, so the second option is correct.
For the third option, there is nothing that proves that these angles are congruent, so the third option is false.
For the fourth option, there is nothing that proves that AC is a diameter, so the fourth option is false.
For the fifth option, the angle ∠BDE inscribes an arc of 180° (semicircle), so it has half the measure of the arc, therefore ∠BDE = 90°. So the fifth option is correct.
she has 78 inches of thread that she cut into 2 pieces. One piece is twice as long as the other piece. How long is each piece?
Answers
Answer:
One piece is 26 inches long while the other is 52 inches long.
We let:
x = one piece of the thread
2x = the other piece of the thread
Since the thread is 78 inches long,
2x + x = 78
Solve for x:
[tex]2x+x=78\Rightarrow3x=78[/tex][tex]\frac{3x}{3}=\frac{78}{3}\Rightarrow x=26[/tex]Since one piece of the thread is 26 inches long, the other piece would be:
[tex]2x=2(26)=52[/tex]The other piece would be 52 inches long.
What is the surface area of the prism below2.5 m2 m10 m1.5 m
Answers
The lateral surface area is the total surface area minus the area of the two triangular faces at the top and bottom of the prism. t
Write an equation in slope-intercept form of the line that passes through the point (5,2) and is parallel to 2y+4x=5
Answers
• We are given 2y +4x = 5
This can be rewritten as :
2y = -4x +5
y = -4/2x +5/2
y = -2x +5/2
• A line parallel to this line will also have the exact, same slope
Therefore m = -2
in order to find the equation , we will use the following formula
y = mx +c , where m = -2 :
y = -2x + c , at point (5;2)
2 = -2(5) +c
2 +10 = c
Therefore c = 12
finally;
y = -2x +12 is our equation .
Every quadrilateral with opposite angles supplementary can be inscribed in a circle.TrueFalseIf a triangle is inscribed in a circle, the center of the circle is called the circumcenter.TrueFalse
Answers
Explanation
The Inscribed Quadrilateral Theorem states that a quadrilateral can be inscribed in a circle if and only if the opposite angles of the quadrilateral are supplementary
Answer 1: True
Also, Given a triangle, the circumscribed circle is the circle that passes through all three vertices of the triangle. The center of the circumscribed circle is the circumcenter of the triangle, the point where the perpendicular bisectors of the sides meet.
Answer 2: True
What is the equation of the graphed function?A. f(x) = -1/3x + 1B. f(x) = 3x + 1C. f(x) = 1/3x + 1D. f(x) = -3x + 1
Answers
If (x,y) is a point in the graph of a line then its coordinates x and y form a solution to the equation of that line. In slope-intercept form this equation looks like this:
[tex]y=mx+b[/tex]What we are going to do here is choose two points from the line in the picture and use them and the expression above to construct two equations for m and b.
As you can see (0,1) and (3,0) are part of the line so we have the following two equations:
[tex]\begin{gathered} 1=m\cdot0+b \\ 0=3m+b \end{gathered}[/tex]From the first equation we get b=1. If we use this value of b in the second equation we obtain the following:
[tex]\begin{gathered} 0=3m+b=3m+1 \\ 0=3m+1 \end{gathered}[/tex]We can substract 1 from both sides:
[tex]\begin{gathered} 0-1=3m+1-1 \\ -1=3m \end{gathered}[/tex]Then we divide both sides by 3:
[tex]\begin{gathered} -\frac{1}{3}=\frac{3m}{3} \\ m=-\frac{1}{3} \end{gathered}[/tex]Then we have this equation for the line in the picture (we take y=f(x)):
[tex]f(x)=-\frac{1}{3}x+1[/tex]AnswerThen the answer is option A.
ZA and ZB are supplementary angles. If mZAZB = (7x – 26)°, then find the measure of ZA.
Answers
the sum of two supplementary angles is 180 degrees
[tex]m\angle A+m\angle B=180[/tex]7x+10+7x-26 = 180
14x = 180 + 16
x = 196/14
x = 14
so the value of m7x+10
= 7(14) + 10
= 98 + 10
= 108
m
Three volunteers are chosen at random from a group of 20 people to help at a camp. How many unique groups of volunteers are possible?
Answers
In mathematics, a combination is a selection of items from a set that has distinct members
Formula
[tex]^n_{^{}}C_r=\frac{n\text{ !}}{(n-r)!r!}[/tex]Where
n = 20
r =3
[tex]\begin{gathered} ^{20}C_3=\frac{20\text{ !}}{(20-3)!3!} \\ \\ \\ ^{20}C_3=\frac{20\text{ !}}{17!3!} \\ \\ \\ ^{20}C_3=\frac{20\text{ }\times19\times18\times17!}{17!3!} \\ ^{20}C_3=\frac{20\text{ }\times19\times18}{3!} \\ \\ ^{20}C_3=\frac{20\text{ }\times19\times18}{3\times2\times1} \\ ^{20}C_3=\frac{20\text{ }\times19\times18}{6} \\ \\ ^{20}C_3=20\text{ }\times19\times3 \\ \\ \\ ^{20}C_3=1140 \end{gathered}[/tex]The final answer
1140 unique groups of volunteers are possible
The total cost function for a product is given by C(x)=3x3−9x2−243x+1229, where x is the number of units produced and C is the cost in hundreds of dollars. Use factoring by grouping and then find the number of units that will give a total cost of at least $50,000. Verify the conclusion with a graphing utility.
Answers
C(x) = 3x³ − 9x² − 243x + 1229
500 = 3x³ − 9x² − 243x + 1229
Subtract 500 on each side, as follows:
0 = 3x³ − 9x² − 243x + 729
(3x³ − 9x²) - (243x - 729) = 0
3x² (x - 3) - 243 (x - 3) = 0
(x - 3)(3x² - 243) = 0
x = 3, x = 9, x = -9; in the context of this problem, where x is the number
of units produced, negative values of x must be omitted, so x = 3 and x = 9
So we can say that if either 3 units or 9 units are produced the total cost
for the product will be at least $50000
[0, 3] U [9,∞)