Final answer:
To find the tangent lines to the curve y = (x - 1)/(x + 1) that are parallel to the given line, convert the given line to slope-intercept form to find the slope, take the derivative of the curve to find where its slope matches the line's slope, and utilize these points to write the equations of the tangent lines.
Explanation:
To find the equations of the tangent lines to the curve y = (x − 1)/(x + 1) that are parallel to the line x − 2y = 3, we first need to find the slope of the given line by rewriting it in slope-intercept form (y = mx + b), where m is the slope. Rewriting x − 2y = 3 gives us y = ⅓x - ⅓; thus, the slope (m) is ⅓.
Next, we find the derivative of the curve, y' = dy/dx, which will give us the slope of the tangent at any point x. Taking the derivative of y = (x − 1)/(x + 1) using the quotient rule or another differentiation method, we find a general expression for y'. We then set y' equal to ⅓ to find the points where the slope of the tangent is equal to the slope of the given line.
After determining the x-values where the tangent has the correct slope, we calculate the corresponding y-values on the curve and use these points to write the equations of the tangent lines in the form y = mx + b, substituting the slope (⅓) and our found points (x, y).
To find the equations of the tangent lines to the given curve that are parallel to the given line, we differentiate the curve's equation to find its slope, equate it to the slope of the given line, solve for x, substitute the values back into the curve's equation to find the corresponding y-values, and use the point-slope form of the equation of a line to find the equations of the tangent lines.
Explanation:To find the equations of the tangent lines to the curve y = (x − 1)/(x + 1) that are parallel to the line x − 2y = 3, we can use the slope of the given line as the slope of the tangent lines. The slope of the given line is 1/2, so the slope of the tangent lines is also 1/2.
Next, we can differentiate the equation of the curve y = (x − 1)/(x + 1) with respect to x to find the slope of the curve at any point. Taking the derivative, we get dy/dx = 2/(x + 1)².
Since the tangent lines are parallel to the given line, their slopes are equal. Therefore, we can equate the slope of the curve to the slope of the tangent lines and solve for x:
2/(x + 1)² = 1/2
Solving this equation, we get x = -1 or x = 1.
Substituting these values of x back into the equation of the curve, we can find the corresponding y-values. The coordinates of the points where the tangent lines intersect the curve are (-1, -2) and (1, 2).
Finally, we can use the point-slope form of the equation of a line to find the equations of the tangent lines:
Tangent line at (-1, -2): y + 2 = (1/2)(x + 1)
Tangent line at (1, 2): y - 2 = (1/2)(x - 1)
Is 3.4 a rational or irrational number? Please explain why, thanks.
Answer:
Yes it is a rational number
I just did it and I passed
Which expression is equal to the number of grams (g) in 2.43 kilograms (kg)?
Answer:
the answer would be D
Step-by-step explanation:
It takes for three gardeners 90 minutes to weed a garden. If the job is to be done in only 15 minutes, how many more gardeners need to be put on the jobs?
Find the slope of the line that passed through points a and d
A) -7/2
B) -2/7
C) 0
D) 2/7
Answer:
The answer is m=2/7 or answer choice D
Step-by-step explanation:
A bus makes 28 stop every 2 hours. how many stops does it make 3 hour? in 4 hours?
A store is having a clearance sale where merchandise on the sales rack is reduced by 80% from the original price. If a jacket was originally priced at $76, what is the sale price.
Farmer tom’s chickens and sheep are all in a jumble. he counts 48 heads and 134 legs, how many sheep does he have?
What value(s) of x make the equation x2 - 18x + 81 = 0 true?
Erica went shopping for new clothes for school. She bought a pair of jeans for $70.19 and several shirts for $8.63 each. If x represents the number of shirts she bought, which of the following equations should be used to find y, the total cost of Erica's shopping trip? Explain your choice.
plzzzz helpppp
At the grocery store, you buy a carton of milk for $1.95, a loaf of bread for $0.85, and a box of cereal for $2.05. Use mental math to find the total cost of the groceries
Which arrangement shows 4.09,4 2/3 ,21/4 and 4.7 in order from least to greatest?
The intensity I of light bulb varies inversely as the square of the distance d from the light bulb. Suppose I is 250 w/m^2 when the distance is 5 m. How much farther would it be to a point where the intensity is 40 w/m^2? ...?
The intensity of a light bulb varies inversely as the square of the distance from the light bulb. To find the distance where the intensity is 40 w/m², use the inverse square law formula to solve for the distance.
Explanation:The intensity of a light bulb varies inversely as the square of the distance from the light bulb. This means that as the distance increases, the intensity decreases.
If the intensity is 250 w/m² at a distance of 5 m, and we want to find the distance where the intensity is 40 w/m², we can use the inverse square law formula:
I₁d₁² = I₂d₂²
Plugging in the values, we get:
250 × (5)² = 40 × (d₂)²
Simplifying, we find that d₂² = 250 × (5)² / 40
d₂² = 312.5
Therefore, d₂ = √312.5 ≈ 17.68
So, it would be approximately 17.68 - 5 = 12.68 meters farther to a point where the intensity is 40 w/m².
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Sure, let's solve it step by step.
First, we know that the initial intensity (I1) is 250 w/m^2 and the initial distance (d1) is 5 m. We are asked to find a distance where the intensity is 40 w/m^2, which would be our final intensity (I2).
According to the inverse square law, the intensity of light is inversely proportional to the square of the distance from the source, i.e., I ∝ 1/d². Therefore, we can say that I1 * d1² = I2 * d2². This equation allows us to find the distance at which the intensity is I2.
So, substituting the given values in the equation, we have 250 * (5)² = 40 * d2².
Now, we can solve for d2. Rearranging terms, d2 = √((250 * (5)²) / 40), which will give us d2 = 12.5 m.
So, the distance at which the intensity of light is 40 w/m^2 is 12.5 m.
However, we have been asked to find the additional distance that needs to be traveled to reach a point where the intensity is 40 w/m^2. This additional distance is the difference between the final distance (d2) and the initial distance (d1), that is, distance_travelled = d2 - d1 = 12.5 - 5 = 7.5 m.
Therefore, you would need to travel an additional 7.5 m to reach a point with an intensity of 40 w/m^2.
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To win at LOTTO in one state, one must correctly select 6 numbers from a collection of 62 numbers (1 through 62). The order in which the selection is made does not matter. How many different selections are possible? ...?
The number of different selections possible when picking 6 numbers from a collection of 62 numbers for a LOTTO game is 62,891,499.
Explanation:The question asks how many different selections are possible when picking 6 numbers from a collection of 62 numbers for a LOTTO game. Since the order of the selection does not matter, we can use the concept of combinations to solve this problem. The formula for combinations is given by C(n, r) = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items selected. Plugging in the values, we have C(62, 6) = 62! / (6! * (62-6)!), which simplifies to 62! / (6! * 56!). Using a calculator or computer program to calculate the factorial, we find that C(62, 6) is equal to 62,891,499.
Final answer:
To find the number of different selections possible in a lotto where 6 numbers are chosen from 62, we use the combination formula: C(62, 6). This yields 45,057,474 possible selections.
Explanation:
Calculation of Different Selections in Lotto
To calculate the number of different selections possible when choosing 6 numbers from a set of 62 without regard to the order of selection, we use the combination formula:
The combination formula is C(n, r) = n! / (r! * (n-r)!), where C(n, r) is the number of combinations, n is the total number of items to choose from, r is the number of items to be chosen, and ! denotes factorial.
In this case, n = 62 and r = 6:
C(62, 6) = 62! / (6! * (62-6)!) = 62! / (6! * 56!)
Calculating the factorials and the division, we find that there are 45,057,474 possible selections.
Therefore, the probability of winning by correctly selecting all 6 numbers is 1 in 45,057,474, making lottery tickets a game of very low probability for individual success.
I need help with subtracting polynomials: (m^2 + 6m - 5) - (7m² - 5m + 1)
A.8m² + m - 4
B.-8m⁴ + 11m - 4
C.-7m² + m - 6
D.-6m² + 11m - 6
To subtract the polynomials, change the signs of the second polynomial and combine like terms. The result is -6m² + 11m - 6, which is option D.
To subtract the given polynomials, we need to subtract the second polynomial from the first one. We do this by changing the signs of the second polynomial and then combining like terms with the first.
(m² + 6m - 5) - (7m² - 5m + 1) becomes (m² + 6m - 5) + (-7m² + 5m - 1).
Now, combine the like terms:
m² - 7m² = -6m²,6m + 5m = 11m,-5 + (-1) = -6.So, the simplified form of the polynomial after subtraction is -6m² + 11m - 6, which corresponds to option D.
What is the missing number in the synthetic-division array?
When the new book Units of Fire was released, Jonathan decided to read it in one sitting for his book report. He started at 11:00 a.m. on Saturday morning and read until 8:00 p.m. that night, for a total of 9 hours. For school, however, he needed to record the exact number of minutes he spent reading.
How many minutes did he read?
...?
The total time in minutes that he spent in reading is:
540 minutes.
Step-by-step explanation:We are asked to find the total number of minutes Jonathan spent on reading on Saturday.
It was given that he read for a total of 9 hours.
Hence, we will convert this time to minutes by using the conversion:
As 1 hour= 60 minutes.
Hence, 9 hours= 9×60 minutes= 540 minutes.
Hence, the total time Jonathan spent in reading on Saturday is:
540 minutes.
whats the difference of 1cm and 1cm2
Final answer:
1 cm is a unit of length equal to [tex]10^{-2}[/tex], meter while [tex]1 cm^2[/tex]is a unit of area equal to 10^{-4} square meters.
Explanation:
The difference between 1 cm and 1 cm2 is that 1 cm is a measure of length, while 1 cm2 is a measure of area. To clarify, 1 cm is equivalent to 10-2 meters (or 0.01 meters), and it represents a one-dimensional measurement. In contrast, 1 cm2 is the area of a square with 1 cm long sides. When converting to square meters (m2), remember that the conversion factor for area is the square of the conversion factor for length. Therefore, 1 cm2 equals (10-2 m)2 or 10-4 m2.
license plates in a particular state display two letters followed by three numbers, such as AT-887 or BB-013. How many different license plates can be manufactured for this state?
The first digit in the quotient of 6523÷7
Thirteen and eight tenths
Jen and Heather went to lunch. When the bill came Jen discovered that she had only 8.00 dollars, luckly Heather had enough money to pay the other part or 3/5 of the bill. ...?
Solve the following equation for B
y=mx+b
Solve the inequality 6x - 8 > 4x + 26
x < 9
x < 17
x > 9
x > 17
Answer:
The Answer is D. x > 17
Hope This Helps!
what is a term without a variable
State the importance of examining a function analytically as well as graphically.?
Answer:
Sometimes it is good to examine function analytically and sometimes it is good to examine function graphically it also depends on the nature of the function.
Moreover, when the function is examined graphically it does not shows the discontinuity of a single point, but it can be examined clearly by analytically.
Also, plotting the graph function is apparently clear by one view but it is not with examining the function analytically.
A liquid-filled glass pyramid is being erected as part of a museum attraction. The pyramid is designed to be 15 feet tall with a rectangular base that is 10 feet long and 6 feet wide. What is the total amount of liquid that the pyramid can hold?
Answer:
300 cubic feet
Step-by-step explanation:
A ball on a string moves around a complete circle, once a second, on a frictionless, horizontal table. The tension in the string is measured to be 6.0N . What would the tension be if the ball went around in only half a second?
If the ball went around in half a second, the tension in the string would be 24.0 N.
Explanation:To find the tension in the string if the ball went around in half a second, we can use the equation for centripetal force. The tension T in the string provides the centripetal force, which is equal to the mass of the ball times the square of its velocity divided by the radius of the circle:
T = (m * v^2) / r
In this case, the mass of the ball and the radius of the circle remain the same, but the velocity changes. Since the ball takes half the time to complete the circle, the velocity is twice as fast. Therefore, the tension in the string would be four times greater:
Tension = 4 * 6.0 = 24.0 N
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What is the lateral surface area of a cylindrical-shaped candle with a radius of 4 cm and a height of 14 cm? (Use 3.14 for pi .)
A.
452.16 sq cm
B.
56 sq cm
C.
351.68 sq cm
D.
1,406.72 sq cm
Answer:
it is c
Step-by-step explanation:
Simplify 659/8. What is the answer rounded to the nearest whole number?
An organic farm uses 3/7 of the land to grow potatoes and 2/5 of the land to grow spinach. The remaining land is used to grow tomato plants. What fraction of the land is used to grow tomato plants?