[tex]\textit{Pythagorean Identities} \\\\ 1+\tan^2(\theta)=\sec^2(\theta)\implies \tan^2(\theta)=\sec^2(\theta)-1 \\\\[-0.35em] ~\dotfill\\\\ \cfrac{\stackrel{ \textit{difference of squares} }{[sec(x)+1][sec(x)-1]}}{sin^2(x)}\implies \cfrac{sec(x)^2-1^2}{sin^2(x)}\implies \cfrac{sec(x)^2-1}{sin^2(x)} \\\\\\ \cfrac{tan^2(x)}{sin^2(x)}\implies \cfrac{sin^2(x)}{cos^2(x)}\cdot \cfrac{1}{sin^2(x)}\implies \cfrac{1}{cos^2(x)}\implies sec^2(x)[/tex]
Coach Walker has a cooler that can hold 5 1/2 gallons of a sports drink. He fills 3/4 of
the cooler with the sports drink to bring to football practice.
B. How many players can fill their water bottles with the sports drink before the cooler is empty? (1 Gallon = 8 pints)
The answer of the question based on the players can fill their water bottles with the sports drink before the cooler is empty is 15 players.
What is Amount?An amount may refer to quantity of something that can be measured, such as amount of substance or amount of the time.
The cooler can hold 5 1/2 gallons of sports drink, and Coach Walker fills 3/4 of it, which means he fills:
5 1/2 gallons * 3/4 = (5 * 2/2 + 1/2) * 3/4 = 15/4 gallons
We know that 1 gallon equals 8 pints, so we can convert the 15/4 gallons to pints:
15/4 gallons * 8 pints/gallon = 30 pints
Therefore, there are 30 pints of sports drink in the cooler. Assuming each player's water bottle can hold 2 pints of sports drink, we can find how many players can fill their bottles before the cooler is empty by dividing the total amount of sports drink by the amount used by each player:
30 pints / 2 pints/player = 15 players
Therefore, 15 players can fill their water bottles with the sports drink before the cooler is empty.
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The box plots display measures from data collected when 20 people were asked about their wait time at a drive-thru restaurant window.
A horizontal line starting at 0, with tick marks every one-half unit up to 32. The line is labeled Wait Time In Minutes. The box extends from 10 to 14.5 on the number line. A line in the box is at 12.5. The lines outside the box end at 5 and 20. The graph is titled Fast Chicken.
A horizontal line starting at 0, with tick marks every one-half unit up to 32. The line is labeled Wait Time In Minutes. The box extends from 8.5 to 15.5 on the number line. A line in the box is at 12. The lines outside the box end at 3 and 27. The graph is titled Super Fast Food.
Which drive-thru is able to estimate their wait time more consistently, and why?
Fast Chicken, because it has a smaller IQR
Fast Chicken, because it has a smaller range
Super Fast Food, because it has a smaller IQR
Super Fast Food, because it has a smaller range
The range, which is the distance between the smallest and greatest linear equation values in the data, does not take into account the distribution of the data within that range.
What is a linear equation?In algebra, a linear equation refers to one with its form y=mx+b. B is the gradient, and m is the esta. The preceding clause is commonly referred to as a "linear function with two variables" so even though y and x are variables. Bivariate linear equations are linear equations with two variables. There are several linear equations: 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, and 3x - y + z = 3. When an equation seems to have the structure y=mx+b, where m is the slope and b is the y-intercept, it is said to be linear. When a measurement seems to have the formula y=mx+b, both with m identifying its slope and b denoting the y-intercept, it is said to be linear.
Because it has a lower IQR, Fast Chicken can more consistently estimate their wait time (Interquartile Range). In this case, the IQR is the range of the middle 50% of the data, which is the distance between the first and third quartiles.
Super Fast Food, on the other hand, has a higher IQR, indicating that the data is more dispersed. This means that customers report a wider range of wait times, making it more difficult to estimate a consistent wait time. The range, which is the distance between the smallest and greatest values in the data, is not as useful for measuring consistency in this case because it does not take into account the distribution of the data within that range.
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What is In 5 + ln 7 + 2ln 6
Answer: The answer to this equation is In (252) + 5.
Step-by-step explanation:
To solve this equation, you will need to first apply the logarithm power identity.
Now evaluate the exponent.
5 + In (7) + In (6^2)
5 + In (7) + In (36)
Apply the logarithm power identity again to the rest of this equation.
5 + In (7) + In (36)
5 + In (7 x 36)
After that, simplify the expression.
-Multiply the numbers & rearrange the terms
5 + In(7 x 36)
In(252) + 5
Therefore, your answer for this equation would be In(252) + 5.
Hope this helps Tekayla!Alyssa Wagner
Middle School Srudent
Which expression does not belong with the other three? explain
The only expression that does not belong to the other 3 is the one that is a binomial.
How to identify the type of algebraic expression?A monomial is defined as an algebraic expression that has only one non- zero term. Examples of a monomial expression are x, 7xy²
A binomial is defined as an algebraic expression that has two non-zero terms. Examples of a binomial expression: x² + 2y
A trinomial is defined as an algebraic expression that possesses three non-zero terms. Examples of a trinomial expression: a + b + c
A polynomial is defined as an algebraic expression that has one, two or more terms.
The only expression that does not belong to the other 3 is (b - 2⁻¹) because it is the only binomial.
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two are supplementary. the measure of one of these angles is 12 degrees less than one-third the measure of the other.what is the measure of each angle
Answer:
two are supplementary. the measure of one of these angles is 12 degrees less than one-third the measure of the other.what is the measure of each angle
Let's call the measures of the two angles x and y, where x is the larger angle. We know that the two angles are supplementary, which means they add up to 180 degrees:
x + y = 180
We also know that one of the angles (let's say y) is 12 degrees less than one-third the measure of the other angle (x):
y = (1/3)x - 12
Now we can substitute the second equation into the first equation to solve for x:
x + (1/3)x - 12 = 180
Multiplying both sides by 3 to get rid of the fraction, we have:
3x + x - 36 = 540
Combining like terms, we get:
4x - 36 = 540
Adding 36 to both sides, we get:
4x = 576
Dividing both sides by 4, we get:
x = 144
Now we can use the first equation to solve for y:
144 + y = 180
Subtracting 144 from both sides, we get:
y = 36
Therefore, the measures of the two angles are 144 degrees and 36 degrees.
Given: QS/TV=RS/UV=3, QR=3TU
Prove: Triangle QRS ~ triangle TUV
A triangle is a geometrical shape that is formed by three straight lines connecting three non-collinear points. The three points are called the vertices of the triangle, and the lines connecting them are called sides or edges.
How to prove that the Triangle QRS ~ triangle TUV?To prove that triangle QRS is similar to triangle TUV, we need to show that their corresponding angles are equal and their corresponding sides are proportional.
Let's start by identifying the corresponding angles:
Angle QRS corresponds to angle TUV (they are both right angles)Angle QSR corresponds to angle TVU (they are opposite angles formed by intersecting lines)Angle RQS corresponds to angle UTV (they are opposite angles formed by intersecting lines)Now, let's look at the corresponding sides:
QS corresponds to TVRS corresponds to UVQR corresponds to TU (given)To prove that the triangles are similar, we need to show that the ratios of the corresponding sides are equal:
QS/TV = RS/UV (given)
QS/TV = 3/3 (since QR=3TU)
QS/TV = 1
RS/UV = 3/3 (since QR=3TU)
RS/UV = 1
Since the ratios of the corresponding sides are equal, the triangles are similar. Therefore, we can conclude that triangle QRS ~ triangle TUV.
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Which of the following equations is the best model for a line of fit for the data?
ŷ = −1.34x + 21.5
ŷ = 1.34x + 21.5
ŷ = −0.75x + 17
ŷ = 0.75x + 17
The correct option is (a) ŷ = -1.34x+21.5 , that is the best model to fit the scatter plot.
What is Scatter plot?A scatter plot (or scatter chart, scatter graph) uses dots to represent values for two(2) different numeric variables. The position of each dot on horizontal and vertical axis indicates values for an individual data point.
The model of the equation best for the plot is,
ŷ = -1.34x+21.5
At point (1,20),
ŷ = -1.34x+21.5
20 = (-1.34 × 1) + 21.5
20 ≈ 20.15
At point (3,19),
ŷ = -1.34x+21.5
19 = (-1.34 × 3) + 21.5
19 ≈17.45
At point (5,15),
ŷ = -1.34x+21.5
15 = (-1.34 × 5) + 21.5
15 ≈ 14.75
The model ŷ = -1.34x+21.5 is giving the best estimation.
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A driver was fined for speeding in 100km/h zone driving 14km in 7m
Calculate the average speed of the car
Answer:
120km/h
Step-by-step explanation:
Distance=14km
Time =7m
A driver was fined for speeding in 100km/h
∴ we convert time into hours
Time =(7/60)h
Average speed of the car = total distance ÷total time
=14km×60/7h=120km/h which is greater than
100km/h
2 ^3 • 2 ^4 is equal to _____.
Answer:
2^7 or 128
Step-by-step explanation:
When we multiply two powers with the same base, we add their exponents. In this case, the base is 2 and the exponents are 3 and 4.
So, 2^3 • 2^4 can be simplified as:
2^3 • 2^4 = 2^(3+4) = 2^7
Therefore, 2^3 • 2^4 is equal to 128.
Answer:128
Step-by-step explanation:
i assume that by ^ you meant power,
2^3=2*2*2=8
2^4=2*2*2*2=16
8*16=128
A deli has two platters of sandwiches. The first platter costs $29 and you get 2 ham sandwiches and 3 turkey sandwiches. The other platter costs $31 and you get 3 ham sandwiches and 2 turkey sandwiches. Let x represent the cost of each ham sandwich and y represent the cost of each turkey sandwich. What is the system of linear equations for the given scenario? What is the cost of each sandwich?
Solution is in da attachment mate!! :D
Step-by-step explanation:
that is a college question ?
x = cost of a ham sandwich
y = cost of a turkey sandwich
2x + 3y = 29
3x + 2y = 31
let's multiply the first equation by 3, abd the second equating by -2, and then we add them :
6x + 9y = 87
-6x - 4y = -62
------------------------
0 5y = 25
y = 25/5 = $5
let's use the original first equation to get x (but we could use also the second equation, it does not matter).
2x + 3×5 = 29
2x + 15 = 29
2x = 14
x = $7
each ham sandwich costs $7.
each turkey sandwich costs $5.
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Which quadrilateral has exactly one pair of parallel sides?
A) rhombus
B) Kite
C) trapezoid
D) parallelogram
What is the result of subtracting 1 and 1/2 from the sum of 4 and 2/3 and 5 and 2/5
Answer:
the result of subtracting 1 and 1/2 from the sum of 4 and 2/3 and 5 and 2/5 is 257/30 = 8 and 39/40.
Step-by-step explanation:
To find the result of subtracting 1 and 1/2 from the sum of 4 and 2/3 and 5 and 2/5, we need to first add the two summands:
4 and 2/3 + 5 and 2/5
To add these mixed numbers, we first need to find a common denominator. The least common multiple of 3 and 5 is 15.
4 and 2/3 can be written as an improper fraction with denominator 3:
4 and 2/3 = 4 x 3/3 + 2/3 = 12/3 + 2/3 = 14/3
5 and 2/5 can be written as an improper fraction with denominator 5:
5 and 2/5 = 5 x 5/5 + 2/5 = 25/5 + 2/5 = 27/5
Now we can add the two fractions with a common denominator of 15:
14/3 + 27/5 = (14 x 5)/(3 x 5) + (27 x 3)/(5 x 3) = 70/15 + 81/15 = 151/15
So, the sum of 4 and 2/3 and 5 and 2/5 is 151/15.
Now we can subtract 1 and 1/2 from this sum:
151/15 - 1 1/2
To subtract mixed numbers, we first need to convert 1 and 1/2 to an improper fraction:
1 and 1/2 = 1 x 2/2 + 1/2 = 2/2 + 1/2 = 3/2
Now we can subtract the fractions with a common denominator of 15:
151/15 - 3/2 = (151 x 2)/(15 x 2) - (3 x 15)/(2 x 15) = 302/30 - 45/30 = 257/30
Therefore, the result of subtracting 1 and 1/2 from the sum of 4 and 2/3 and 5 and 2/5 is 257/30.
A certain virus infects one in every 300 people. A test used to detect the virus in a person is positive
90% of the time if the person has the virus and 5% of the time if the person does not have the virus.
(This 5% result is called a false positive.) Let A be the event "the person is infected" and B be the
event "the person tests positive",
a) Find the probability that a person has the virus given that they have tested positive, i.e. find
P(A|B). Round your answer to the nearest tenth of a percent and do not include a percent sign.
P(A|B)=
%
b) Find the probability that a person does not have the virus given that they test negative, i.e. find
P(A'|B'). Round your answer to the nearest tenth of a percent and do not include a percent sign.
P(A'|B')=
Answer:
b i think let me know if im right
Step-by-step explanation:
simplify sqrt (196y^8)
this is basically saying the square root of 196y to the power of 8.
btw i can do this. i just feel rlly lazy today :[
Hence, in response to the provided question, we can say that As a result, exponents [tex]/sqrt(196y^8)[/tex] simplifies to [tex]14y^4[/tex].
what are exponents?Exponentiation, sometimes known as "raising b to the nth power," is a substitution cipher that includes two numbers: a base quantity, b, and an argument, meaning power, n. Exponents represent the number of times a value has been increased by itself. As an example, 2-3 (represented as 23) represents: 2 x 2 x 2 = 8. 2 + 3 = 6 does not equal 23. Remember that an integer multiplied by one equals itself. Exponents are a method for expressing large numbers as powers. The amount that represents however many times a statistic has already been amplified by itself is called the exponent. As an example, multiplying 6 by 4 yields 6 x 6 x 6.
Using exponent principles and the square root of perfect squares, we may simplify sqrt(196y8) as follows:
[tex]\sqrt(196y8) = \sqrt((142*(y4)2) [since 196 = 142 and (y4)2 = y8]\\\sqrt(142) * \sqrt((y4)2) [using the \sqrt(ab) = \sqrt(a) * \sqrt(b) rule]\\= 14y^4[/tex]
As a result, [tex]/sqrt(196y^8)[/tex] simplifies to [tex]14y^4[/tex].
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9 + 6g + 1 = 100 please I need help these are difficult
Answer:
So g would be 15.
will the product of 2 numbers increase or decraese and by what percent if one of them is increased by 50% and the other one is decraesed by 50%
Answer: Let's assume the two original numbers to be x and y.
If one of them is increased by 50%, then the new value will be 1.5x, and if the other one is decreased by 50%, the new value will be 0.5y.
The product of the two new numbers will be:
1.5x * 0.5y = 0.75xy
So, the new product is 0.75 times the original product. This means the product of the two numbers has decreased by 25%.
To summarize:
If one number is increased by 50% and the other number is decreased by 50%, the product of the two numbers will decrease by 25%.
The new product is 0.75 times the original product.
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In a science experiment the temperature of a liquid is first read at 11.15am. If the temperature is read every 12 hours, at what time will the eight temperature reading be taken?
If the temperature is read every 12 hours, then the eighth temperature reading will be taken at 11.15 pm three days after the first reading.
What is temperature?Temperature is a measure of the degree of hotness or coldness of a substance or object, typically measured using a thermometer in degrees Celsius or Fahrenheit.
If the temperature is read every 12 hours, then the time interval between two consecutive temperature readings is 12 hours.
To find the time of the eighth temperature reading, we need to add 12 hours for each of the previous seven temperature readings:
First reading: 11.15 am
Second reading: 11.15 am + 12 hours = 11.15 pm
Third reading: 11.15 pm + 12 hours = 11.15 am (next day)
Fourth reading: 11.15 am (next day) + 12 hours = 11.15 pm (next day)
Fifth reading: 11.15 pm (next day) + 12 hours = 11.15 am (two days after the first reading)
Sixth reading: 11.15 am (two days after the first reading) + 12 hours = 11.15 pm (two days after the first reading)
Seventh reading: 11.15 pm (two days after the first reading) + 12 hours = 11.15 am (three days after the first reading)
Eighth reading: 11.15 am (three days after the first reading) + 12 hours =
11.15 pm (three days after the first reading)
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A town's population is currently 20,000. If the population doubles every 34 years, what will
the population be 68 years from now?
Answer:
Here, given
present population = 20000
according to question,
population gets double every 34 years
i.e P = 20000 × 2
= 40000
now,
P = 40000
T = 68-34 = 34
ie, again p is double
so, P = 40000×2
= 80000
Hence, the population after 68 years from now is 80000
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What is the area of a triangle with a base of 23 feet and a height of 6 feet?
A) 26 ft2
B) 58 ft2
C) 69 ft2
D) 138 ft2
Answer:
C) 69
Step-by-step explanation:
23 x 6 ÷ 2 =69feet2
ok!
Megan flies a drone in a circular path around an object that is 180 feet west and 180 feet north of her position. The drone's path takes it over a point that is 220 feet east and 230 feet south of her.
Find an equation for the drone's path. (Assume Megan is located at the origin, with the horizontal axis running east-west and the vertical axis running north-south)
The drone's path follows the equation __________
When the drone passes due north of Megan's position, it will be ___________ feet north of her (round your answer to three decimal places).
To find the equation of the drone's path, we can first find the coordinates of the center of the circle that the drone is flying around. We can do this by finding the midpoint between the two points the drone passes over:
Midpoint in the x-direction: (220 ft - 180 ft)/2 = 20 ft to the right of the origin.
Midpoint in the y-direction: (230 ft - 180 ft)/2 = 25 ft above the origin.
Therefore, the center of the circle is located at (20, 25) ft.
The radius of the circle can be found by calculating the distance between the center of the circle and either of the two points the drone passes over:
Radius: sqrt((20-(-180))^2 + (25-180)^2) = sqrt(40000 + 15625) = 205 ft (rounded to the nearest whole number)
So the equation for the drone's path is:
(x - 20)^2 + (y - 25)^2 = 205^2
To find how far north of Megan's position the drone is when it passes due north, we can substitute x = 0 into the equation:
(0 - 20)^2 + (y - 25)^2 = 205^2
400 + (y - 25)^2 = 42025
(y - 25)^2 = 41625
y - 25 = +/-sqrt(41625)
y = 25 +/- 204.06
So the drone is either 229.06 ft north or 22.94 ft south of Megan's position when it passes due north. Rounded to three decimal places, the answer is 229.06 ft north.
i need the answer for this question
The value of the functions are;
f(3) = - 3
g(5) = -77
What is a function?A function can be defined as an equation or expression hat shows the relationship between two variables.
These variables are termed;
The dependent variableThe independent variableFrom the information given, we have that;
f(x) = -5x + 2
g(x) = -3x²- 2
To determine the function, f(3), and g(5), we have to substitute the value of x as 3 in the function f(x) and the value of x as 5 in the function g(x).
We have;
f(3) = -5(3) + 2
expand the bracket
f(3) = - 13
For the second function;
g(5) = - 3(5)² -2
g(5) = -77
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Multiply.
4 1/3 x 2 3/4
7 1/12
8 1/4
11 11/12
i dont know
Answer:
11 11/12
Step-by-step explanation:
You can change these "mixed numbers" (a big whole numbers and also a fraction) to "improper fractions" (a single fraction with a bigger number on top and a smaller number on the bottom)
4 1/3 × 2 3/4
see image
13/3 × 11/4
Multiply straight across, top×top and bottom×bottom.
see image.
Change back to a mixed number by dividing.
The multiplication of 4 1/3 x 2 3/4 is 11 11/12.
The correct option is C.
What is an improper fraction?A fraction that has the numerator higher than or equal to the denominator is said to be an improper fraction.
To multiply 4 1/3 and 2 3/4, we can first convert them to improper fractions:
4 1/3 = 13/3
2 3/4 = 11/4
Then we can multiply the fractions by multiplying the numerators and denominators separately:
(13/3) x (11/4) = (143/12)
Finally, we can convert the improper fraction back to a mixed number if desired:
143/12 = 11 11/12
Therefore, the answer is 11 11/12.
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At Cheng's Bike Rentals, it costs $22 to rent a bike for 4 hours. How many dollars does it cost per hour of bike use?
Answer:
Step-by-step explanation:
To find the cost per hour of bike use at Cheng's Bike Rentals, we need to divide the total cost of renting the bike by the number of hours it was rented for.
If it costs $22 to rent a bike for 4 hours, then the cost per hour is:
$22 / 4 hours = $5.50/hour
Therefore, it costs $5.50 per hour of bike use at Cheng's Bike Rentals.
Find the ordered pair solutions for the system of equations. ([?], f(x) = x² + 1 f(x) = -x + 3 ) and ( Enter the smallest x first.
The ordered pair solutions for the system of equations are (-2, 5) and (1, 2).
How to determine ordered pair?To determine if an ordered pair is a solution to two systems of equations, substitute the values of the variables into each equation. If an ordered pair makes both equations true, it is the solution of the system.
To find the ordered pair solutions for the system of equations, we need to solve the two equations simultaneously.
f(x) = x² + 1 ...(1)
f(x) = -x + 3 ...(2)
Setting the two equations equal to each other, we get:
x² + 1 = -x + 3
Rearranging this equation, we get:
x² + x - 2 = 0
Factoring this quadratic equation, we get:
(x + 2)(x - 1) = 0
Therefore, the solutions for x are x = -2 and x = 1.
Substituting these values of x into either equation (1) or (2), we get:
For x = -2: f(-2) = (-2)² + 1 = 5, and f(-2) = -(-2) + 3 = 5.
For x = 1: f(1) = 1² + 1 = 2, and f(1) = -1 + 3 = 2.
Therefore, the ordered pair solutions for the system of equations are (-2, 5) and (1, 2).
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In ΔQRS, q = 3.9 cm, � m∠S=10° and � m∠Q=74°. Find the length of s, to the nearest 10th of a centimeter.
S thus measures around 2.77 centimetres in length.
What is the purpose of law of sines?The law of sines is frequently used to find the elusive side or angle of a triangle. This law can be used if precise triangle measurement combinations are given. ASA The objective is to identify the unknown side given two angles and an included side.
The Law of Sines can be used to determine the length of side s: s/sin(mS) = q/sin(mQ).
replacing the specified values:
s/sin(10°) = 3.9/sin(74°)
s ≈ sin(10°) × 3.9 ÷ sin(74°)
s ≈ 0.684 × 3.9 ÷ 0.961
s ≈ 2.77 cm (rounded to the nearest 10th)
S thus measures around 2.77 centimetres in length.
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CAN SOMEONE HELP WITH THIS QUESTION?
Answers:
[tex]\text{Derivative: } \ \ \frac{dy}{dx} = \frac{-260x^{9} - 156x^{25}y}{6x^{26}+7y^6}\\\\ \text{Tangent line at (1,1) is: } \ y = -32x + 33\\\\[/tex]
==========================================================
Work Shown:
Let's determine the derivative dy/dx.
Part 1
[tex]26x^{10} + 6x^{26}y+y^7 = 33\\\\ \frac{d}{dx}(26x^{10} + 6x^{26}y+y^7) = \frac{d}{dx}(33)\\\\ \frac{d}{dx}(26x^{10}) + \frac{d}{dx}(6x^{26}y)+\frac{d}{dx}(y^7) = 0\\\\ 10*26x^{10-1} + \frac{d}{dx}(6x^{26})y+(6x^{26})*\frac{dy}{dx}+7y^6\frac{dy}{dx} = 0\\\\[/tex]
Part 2
[tex]260x^{9} + 26*6x^{26-1}y+6x^{26}*\frac{dy}{dx}+7y^6\frac{dy}{dx} = 0\\\\ 260x^{9} + 156x^{25}y+6x^{26}*\frac{dy}{dx}+7y^6\frac{dy}{dx} = 0\\\\ 260x^{9} + 156x^{25}y+(6x^{26}+7y^6)\frac{dy}{dx} = 0\\\\ (6x^{26}+7y^6)\frac{dy}{dx} = -260x^{9} - 156x^{25}y\\\\ \frac{dy}{dx} = \frac{-260x^{9} - 156x^{25}y}{6x^{26}+7y^6}\\\\[/tex]
There are many other possible ways to express the dy/dx expression.
GeoGebra and WolframAlpha are two useful tools to help verify the answer. Make sure you use the CAS mode in GeoGebra.
-------------------------------------------
Part 3
Now that we know dy/dx, we can determine the slope of the tangent at any point (x,y) on the implicit function curve.
Plug in x = 1 and y = 1.
[tex]\frac{dy}{dx} = \frac{-260x^{9} - 156x^{25}y}{6x^{26}+7y^6}\\\\ \frac{dy}{dx} = \frac{-260(1)^{9} - 156(1)^{25}(1)}{6(1)^{26}+7(1)^6}\\\\ \frac{dy}{dx} = \frac{-260(1) - 156(1)(1)}{6(1)+7(1)}\\\\ \frac{dy}{dx} = \frac{-260 - 156}{6+7}\\\\ \frac{dy}{dx} = \frac{-416}{13}\\\\ \frac{dy}{dx} = -32\\\\[/tex]
The slope of the tangent line at (1,1) is m = -32.
-------------------------------------------
Part 4
Apply the point-slope formula to determine the tangent line.
[tex]m = -32 = \text{ slope}\\(x_1,y_1) = (1,1) = \text{the point the tangent line goes through}[/tex]
So,
[tex]y - y_1 = m(x - x_1)\\\\y - 1 = -32(x - 1)\\\\y - 1 = -32x + 32\\\\y = -32x + 32 + 1\\\\y = -32x + 33\\\\[/tex]
algebra 1
what is the written formula for standard form
Answer: Ax+By=C This is the formula
Determine the equation of the hyperbola with foci (3, 11) and (3, -9), and co-
vertices (11, 1) and (-5,1).
The equation of the hyperbola with the given foci and vertices is (x - 3)² - (y - 1)² = 64
What is hyperbola?Hyperbola is a type of conic section, which is a curve formed by the intersection of a cone with a plane. It is similar to a circle, but with two separate halves that are mirror images of each other. These two halves are called branches, and the point where they intersect is the vertex. The hyperbola is characterized by its two foci, which are points that lie on the inside of the structure.
The equation of a hyperbola with foci (h, k) and (h, l) and vertices (m, n) and (p, n) is given by:
((x - h)² / (m - h)²) - ((y - n)² / (k - n)²) = 1
In this case, h = 3, k = 11, l = -9, m = 11, and n = 1. Plugging these values into the equation, we get:
((x - 3)² / (11 - 3)²) - ((y - 1)² / (11 - 1)²) = 1
Simplifying, we get:
(x - 3)² - (y - 1)² = 8²
Therefore, the equation of the hyperbola with the given foci and vertices is: (x - 3)² - (y - 1)² = 64.
To learn more about hyperbola
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Solve the following system of equations:
y = 3x + 1
y = 3x + 7
(3,1)
(3,7)
No Solutions
Infinite Solutions
Answer:
No solutions
Step-by-step explanation:
i have a feeling it's "no solutions". looking at the equations, the only thing that's different is the y intercepts (one being +1 and the other +7). that tells me they are parallel, so they'll never touch each other
8.
A right triangle shaped sail has an area of 150 square
meters. The base of the sail is 10 less than twice the
ight. Find the base and the height.
Answer:
Base=20m height= 15m
Step-by-step explanation:
The area of a triangle is given by:
[tex]A=\frac{bh}{2}[/tex]
since base is 10 less than twice the height b=2h-10
plugin in those values and knowing area is 150
[tex]150=[/tex][tex]\frac{h(2h-10)}{2}[/tex]
then solve for h
[tex]300=2h^2-10h[/tex] this is quadratic equation
[tex]h^2-5h-150=0[/tex]
factorizing (notice you can also use quadratic equation)
[tex](h-15)(h+10)=0[/tex]
which positive solution (height cant be negative) is h=15
then the base is b=2(15)-10=20
Answer:
The height is 15 meters and the base is 20 meters.
Step-by-step explanation:
Let's use the formula for the area of a right triangle:
A = (1/2)bh
Where A is the area, b is the base, and h is the height.
We're given that the area is 150 square meters, so we can substitute that in:
150 = (1/2)bh
Next, we're told that the base is 10 less than twice the height. In other words,
b = 2h - 10
We can substitute this expression for b into the equation for the area:
150 = (1/2)(2h - 10)h
Simplifying:
300 = (2h - 10)h
300 = 2h^2 - 10h
2h^2 - 10h - 300 = 0
Dividing both sides by 2:
h^2 - 5h - 150 = 0
Now we can solve for h using the quadratic formula:
h = (-(-5) ± sqrt((-5)^2 - 4(1)(-150))) / 2(1)
h = (5 ± sqrt(625)) / 2
h = (5 ± 25) / 2
We can ignore the negative root (which gives us a negative height), so:
h = 15
Now we can use the expression for b in terms of h to find the base:
b = 2h - 10
b = 2(15) - 10
b = 20
Therefore, the height is 15 meters and the base is 20 meters.