The tangent line is the point that touches a graph at a point.
The value of x at the tangent line to the graph of [tex]\mathbf{y=-9x^2 - 2x}[/tex] is [tex]\mathbf{x = -\frac 19 }[/tex]
The function is given as:
[tex]\mathbf{y=-9x^2 - 2x}[/tex]
Differentiate both sides with respect to x
[tex]\mathbf{y' =-18x - 2}[/tex]
Set the above equation to 0, to calculate the value of x
[tex]\mathbf{-18x - 2 = 0}[/tex]
Collect like terms
[tex]\mathbf{-18x = 2 }[/tex]
Divide both sides by -18
[tex]\mathbf{x = -\frac 19 }[/tex]
Hence, the value of x when at tangent line to the graph of [tex]\mathbf{y=-9x^2 - 2x}[/tex] is [tex]\mathbf{x = -\frac 19 }[/tex]
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What does it mean to say that lim n → ∞ an = ∞? the terms an become small as n becomes large. the terms an approach zero as n becomes large. the terms an become large as n becomes large. the terms an become small as n becomes small. the terms an become large as n becomes small?
The limit [tex]\lim_{n \to \infty} a_n = \infty[/tex] means that the term [tex]a_n[/tex] becomes large as n becomes large
The limit of a function describes the behavior of such function as the independent variable approaches a certain value
Infinity [tex]\infty[/tex] represents a very large number
In the limit, both n and [tex]a_n[/tex] tend to infinity
The given limit is:
[tex]\lim_{n \to \infty} a_n = \infty[/tex]
The limit above describes the behavior of [tex]a_n[/tex] as n approaches infinity
As n approaches infinity [tex]a_n[/tex] also approaches infinity
Therefore, the description of the limit [tex]\lim_{n \to \infty} a_n = \infty[/tex] is that the term [tex]a_n[/tex] becomes large as n becomes large
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The statement 'lim n → ∞ an = ∞' means that the terms an become large as n becomes large.
Explanation:The statement lim n → ∞ an = ∞ means that the terms an become large as n becomes large.
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The base of a triangle exceeds the height by 5 inches. If the area is 75 square inches, find the length of the base and the height of the triangle
Final answer:
To find the base and height of the triangle with an area of 75 square inches and the base exceeding the height by 5 inches, we set up a quadratic equation from the area formula A = (bh)/2. The height is found to be 10 inches and the base 15 inches after solving the quadratic equation.
Explanation:
To solve for the base and height of a triangle given the area, we can use the formula for the area of a triangle which is A = (bh)/2, where A is the area, b is the base and h is the height. Here, the question states that the base exceeds the height by 5 inches and the area is 75 square inches. Let's define h as the height and then the base will be h + 5. Substituting the given values into the area formula, we get:
75 = (h + 5)h/2
By multiplying both sides by 2 to eliminate the fraction, we get:
150 = h(h + 5)
This expands to a quadratic equation:
h² + 5h - 150 = 0
Solving for h using the quadratic formula, the positive solution is the height of the triangle:
h = (-5 + √(5² + 4·150))/2
h = (-5 + √625)/2
h = (-5 + 25)/2
h = 20/2
h = 10 inches. Therefore, the base is:
b = 10 + 5
b = 15 inches.
The height of the triangle is 10 inches, and the base is 15 inches.
Write the rate as a unit rate
A Jockey had 1053 wins in 31 years of riding
The unit rate is win per year
Let f(x) = 3x+2 and g(x) = 7x+6. find f×x
What is the probability that x is between $34 and $38?
The probability that ( x) is between $34 and $38 is 0.6826 or 68.26%.
To find the probability that a random variable ( x) is between $34 and $38, we need to use the probability density function (PDF) if ( x) follows a continuous distribution. Let's assume ( x ) follows a normal distribution with a mean [tex](\( \mu \))[/tex] of $36 and a standard deviation [tex](\( \sigma \))[/tex] of $2.
Calculate the z-scores for $34 and $38:
[tex]\[ z_1 = \frac{34 - 36}{2} = -1 \][/tex]
[tex]\[ z_2 = \frac{38 - 36}{2} = 1 \][/tex]
Look up the z-scores in the standard normal distribution table:
[tex]\[ P(z_1 < Z < z_2) = P(-1 < Z < 1) \][/tex]
Find the probabilities corresponding to these z-scores:
[tex]\[ P(Z < 1) = 0.8413 \][/tex]
[tex]\[ P(Z < -1) = 0.1587 \][/tex]
Subtract the two probabilities:
[tex]\[ P(-1 < Z < 1) = P(Z < 1) - P(Z < -1) = 0.8413 - 0.1587 = 0.6826 \][/tex]
So, the probability that ( x) is between $34 and $38 is 0.6826 or 68.26%.
Selena babysits on the weekends. The equation y = 12x represents the amount of money she earns for every hour, x, she babysits. What is the constant of proportionality?
I REALLY NEED HELP WITH MY CALC CORRECTIONS. THIS IS THE THIRD TIME IVE POSTED THIS. I NEED HELP ASAP PLS!! I HATE CALC 1
Sarah polled 40 randomly selected students at her high school and found that 20% ( = 0.2) are happy with the quality of the cafeteria food. With a desired confidence level of 99%, which has a corresponding z*-score of 2.58, what is the approximate margin of error of Sarah’s poll? Remember, the margin of error, E, can be determined using the formula E = z*
A hair salon charges a fixed rate of $25.00 for a haircut and then an additional $15.00 for any other services. Write a function to model the cost of services there and then evaluate it if you also get a shampoo, highlight, and blow dry.
Final answer:
The cost function for the salon is C(n) = 25 + 15n, where n is the number of additional services. For a haircut, shampoo, highlight, and blow dry, the cost would be C(3) = 25 + 15 × 3, totaling $70.
Explanation:
The cost of services at the hair salon can be modeled by the function C(n) = 25 + 15n, where C is the total cost in dollars and n is the number of additional services.
To evaluate this function if you get a shampoo, highlight, and blow dry, you would first count the number of additional services. In this case, there are three additional services.
So, C(3) would equal 25 + 15 × 3 which is 25 + 45 or $70.
The cost for a haircut, shampoo, highlight, and blow dry would be $70.
A father is three times as old as his son, and the sum of their age is 76 years old. How old is the son?
Age of son = 19 years
Age of father = 57 years
What is ratio?The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity.
Given,
A father is three times as old as his son
Sum of their age is 76 years
Let age of son is x, age of father = 3x
x + 3x = 76
4x = 76
x = 76/4
x = 19
Age of son = 19 years
Age of father = 3×19 = 57 years
Hence, 19 years is the age of son and
57 years is the age of father.
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The weight of 1000 identical samples of a substance is 10 pounds. What is the weight of 10 samples?
2/3 x 5/8 in simplest form
Determine whether or not f is a conservative vector field. if it is, find a function f such that f = ∇f. if it is not, enter none. f(x, y) = (yex + sin y) i + (ex + x cos y) j
The vector field f(x, y) = (yex + sin y) i + (ex + x cos y) j is conservative because its curl vanishes everywhere. The function F such that ∇F = f is yex + cos y + C.
Explanation:To determine if the given function f(x, y) = (yex + sin y) i + (ex + x cos y) j is a conservative vector field, we need to see if the curl of the vector field vanishes everywhere. The curl of a vector field in two dimensions is the partial derivative of the j component with respect to x minus the partial derivative of the i component with respect to y.
For f(x, y), the partial derivative of the j component (ex + x cos y) with respect to x is ex + cos y, and the partial derivative of the i component (yex + sin y) with respect to y is ex + cos y. Both derivatives are equal, implying the curl is zero. Hence, f(x, y) is a conservative vector field.
Next, to find a function F such that ∇F = f, we need to integrate. The potential function, F, can be obtained by integrating the i component of f with respect to x (since it does not involve y), and the j component of f with respect to y (since it does not involve x): ∫yex dx + ∫sin y dy = yex + cos y + C. This is the function F such that ∇F = f.
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During the World Series the host city wide hotel occupancy rose 21% over the prior busiest night. If the busiest night had been 76% occupany, what was the new record? (use % symbol in answer)
Answer:
91.96%.
Step-by-step explanation:
We have been given that during the World Series the host city wide hotel occupancy rose 21% over the prior busiest night. The busiest night had been 76% occupancy.
[tex]\text{New record}=\text{Old record+ Increase in old record}[/tex]
To find the new record of occupancy, we will find 21% of 76% and add it to 76%.
[tex]\text{The new record}=76\%+({21\%\text{ of } 76\%)[/tex]
[tex]\text{The new record}=\frac{76}{100}+({\frac{21}{100}\times\frac{76}{100})[/tex]
[tex]\text{The new record}=0.76+({0.21\times 0.76)[/tex]
[tex]\text{The new record}=0.76+0.1596[/tex]
[tex]\text{The new record}=0.9196[/tex]
Now let us convert our answer in percentage by multiplying our answer by 100.
[tex]\text{The new record}=0.9196\times 100=91.96\%[/tex]
Therefore, the new record of occupancy was 91.96%.
Divide and give the answer in scientific notation.
8.4 x 10^7
----------------
3 x 10^3
Let a and b be events in a sample space s such that p(a) = 0.41, p(b) = 0.46 and p(a ∩
b.= 0.14. find p(a | b).
The conditional probability that event a will occur given that event b has occurred, denoted as p(a | b), is approximately 0.3 or 30%.
Explanation:The question is asking for the conditional probability of the event a given that event b has occurred, written as p(a | b). In probability theory, the formula for conditional probability is p(a | b) = p(a ∩ b) / p(b). Here, we already know that p(a ∩ b) = 0.14 and p(b) = 0.46. Substituting these values into the formula gives us:
p(a | b) = 0.14 / 0.46 = 0.304347826.
So, the probability of event a occurring given that event b has occurred is approximately 0.3 or 30%.
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which of the following represents the factorization of trinomal below
What you put on social media matters. Why does it matter and how will you ensure your social media postings reflect you professionally?
Answer:
Step-by-step explanation:
Great question, it is always good to ask away and get rid of any doubts that you may be having.
As a professional what you put on social media matters a lot. Social Media postings show all what you are thinking or doing to hundreds, thousands, or even millions of people. If you are an artist you depend on those people (fans) to continue supporting you and your art so a single wrong post can turn a lot of people against you.
If you work for a company you are not only representing yourself through your posts but the company as well. This is because a company hires you for the person that you are therefore it is seen through the public eye that they share your way of thinking.
The best way for your postings to reflect you professionally, is to post only things relating to your own career and journey. While leaving out any opinions, personal information, and controversial topics.
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If f(x) is the total number of bacteria present in a sample after x hours , which of the following statements best describes the meaning of f(2)=12
A. The total number of bacteria present in a sample after 12 hours is 2.
B. The total number of bacteria present in the sample after 6 hours is 12.
C. The total number of bacteria present in the sample after 2 hours is 12.
D. The total number of bacteria present in the sample after 6 hours is 12.
Answer: C. The total number of bacteria present in the sample after 2 hours is 12.
Step-by-step explanation:
The given statement : f(x) is the total number of bacteria present in a sample after x hours.
It means the independent quantity = Number of hours
i.e. The independent variable in the given situation = x
Also, the dependent quantity = Total number of bacteria
i.e. The dependent variable in the given situation = f(x)
Therefore, the meaning of "f(2)=12" is the total number of bacteris present in sample is 12 after 2 hours.
A family paid 23,100 as a down payment for a home if this represents 14% of the price of the home find the price of the home
find the percent of each number 30% of 35
Jane saved $ 800. Her sister has 10 times as much money. How much money does Jane's sister have? Use number or words to explain your answer.
The perimeter of a college athletic field is 104 meters and the length is 16m more than the width find the length and width
The length and width of the rectangle will be 19 meters.
What is the perimeter of the rectangle?The sum of all the sides of the rectangle will be known as the perimeter of the rectangle. Let L be the length and W be the width of the rectangle.
Then the perimeter of the rectangle will be
Perimeter of the rectangle = 2(L + W) units
The perimeter of a college athletic field is 104 meters and the length is 16 meters more than the width.
L = 14 + W
Then the length and width of the rectangle will be
P = 2(L + W)
104 = 2(14 + W + W)
52 = 14 + 2W
2W = 38
W = 19 meters
The length and width of the rectangle will be 19 meters.
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A scientist placed 22.854 grams of sodium chloride in a beaker. He then added five times as much calcium chloride as sodium chloride. Finally, he divided the mixture evenly into four dishes.
How much of the mixture did each dish contain?
A .28.5675 g
B. 34.2810 g
C. 137.1240 g
D. 548.4960 g
The value of n is both 5 times as much as the value of m and 36 more than the value of m. What are the values of n and m? Explain.
Which of the following terms best describes the graph of the exponential function given below? F(x)=9*((1)/(7))^(x)
A. Linear
B.Decreasing
C.Increasing
D.Quadratic
Answer:
B. Decreasing
Step-by-step explanation:
We have been given formula of a function [tex]F(x)=9\cdot (\frac{1}{7})^x[/tex]. We are supposed to determine the type of our given function.
We know that an exponential function is in form [tex]y=a\cdot b^x[/tex], where,
a = Initial value,
b = If [tex]b>1[/tex], then the function is growth function. If [tex]b<1[/tex], then the function is decay function.
Upon looking at our given function we can see it is in form of exponential function.
Upon comparing our given function with standard exponential function, we can see that the value of a is 9 and value of b is [tex]\frac{1}{7}[/tex].
Since [tex]\frac{1}{7}<1[/tex], therefore, our function is decreasing and option B is the correct choice.
If rectangle ABCD is resized with a central point (−1, 1) and scale factor 3, what are the coordinates of D'?
A)(−3, 3)B)(−10, −5)C)(−5, −10)D)(−15, −15)
Find an angle between 0° and 360° that is coterminal with 456°.
rewrite y^2(9y^2+4y-9) in standard form
How do I figure this out?