Find the slope m and an equation of the tangent line to the graph of the function f at the specified point. (Simplify your answer completely.) f(x) Slope: -13/49 Equation: = x + 3 x² + 3 (2,5/7) (Give your answer in the slope-intercept form.)
The number of bacteria N(t) in a certain culture t min after an experimental bactericide is introduced is given by 9400 1 + t² (a) Find the rate of change of the number of bacteria in the culture 3 min after the bactericide is introduced. bacteria/min N(t) = + 1600 (b) What is the population of the bacteria in the culture 3 min after the bactericide is introduced? bacteria
Answers
The slope of the tangent line to the graph of the function f(x) = x + 3x² + 3 at the point (2, 5/7) is -13/49. The equation of the tangent line can be written in the slope-intercept form as y = (-13/49)x + 41/49.
To find the slope of the tangent line, we need to find the derivative of the function f(x) = x + 3x² + 3 and evaluate it at x = 2. Taking the derivative, we have:
f'(x) = 1 + 6x.
Evaluating f'(x) at x = 2, we get:
f'(2) = 1 + 6(2) = 1 + 12 = 13.
Therefore, the slope of the tangent line at the point (2, 5/7) is 13.
To find the equation of the tangent line, we use the point-slope form:
y - y₁ = m(x - x₁),
where (x₁, y₁) is the given point and m is the slope. Plugging in the values, we have:
y - 5/7 = (-13/49)(x - 2).
Simplifying, we get:
y - 5/7 = (-13/49)x + 26/49,
y = (-13/49)x + 41/49.
Therefore, the equation of the tangent line to the graph of f at the point (2, 5/7) is y = (-13/49)x + 41/49.
Moving on to the second question, we are given the function N(t) = 9400/(1 + t²), which represents the number of bacteria in the culture t minutes after the bactericide is introduced.
(a) To find the rate of change of the number of bacteria in the culture 3 minutes after the bactericide is introduced, we need to find the derivative N'(t) and evaluate it at t = 3. Taking the derivative, we have:
N'(t) = -9400(2t)/(1 + t²)².
Evaluating N'(t) at t = 3, we get:
N'(3) = -9400(2(3))/(1 + 3²)² = -9400(6)/(1 + 9)² = -9400(6)/10² = -9400(6)/100 = -5640.
Therefore, the rate of change of the number of bacteria in the culture 3 minutes after the bactericide is introduced is -5640 bacteria/min.
(b) To find the population of the bacteria in the culture 3 minutes after the bactericide is introduced, we plug in t = 3 into the function N(t):
N(3) = 9400/(1 + 3²) = 9400/(1 + 9) = 9400/10 = 940.
Therefore, the population of the bacteria in the culture 3 minutes after the bactericide is introduced is 940 bacteria.
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The population of the bacteria in the culture 3 minutes after the bactericide is introduced is 940 bacteria. The rate of change of the number of bacteria in the culture 3 minutes after the bactericide is introduced is -5640 bacteria/min.
The slope of the tangent line to the graph of the function f(x) = x + 3x² + 3 at the point (2, 5/7) is -13/49. The equation of the tangent line can be written in the slope-intercept form as y = (-13/49)x + 41/49.
To find the slope of the tangent line, we need to find the derivative of the function f(x) = x + 3x² + 3 and evaluate it at x = 2. Taking the derivative, we have:
f'(x) = 1 + 6x.
Evaluating f'(x) at x = 2, we get:
f'(2) = 1 + 6(2) = 1 + 12 = 13.
Therefore, the slope of the tangent line at the point (2, 5/7) is 13.
To find the equation of the tangent line, we use the point-slope form:
y - y₁ = m(x - x₁),
where (x₁, y₁) is the given point and m is the slope. Plugging in the values, we have:
y - 5/7 = (-13/49)(x - 2).
Simplifying, we get:
y - 5/7 = (-13/49)x + 26/49,
y = (-13/49)x + 41/49.
Therefore, the equation of the tangent line to the graph of f at the point (2, 5/7) is y = (-13/49)x + 41/49.
Moving on to the second question, we are given the function N(t) = 9400/(1 + t²), which represents the number of bacteria in the culture t minutes after the bactericide is introduced.
(a) To find the rate of change of the number of bacteria in the culture 3 minutes after the bactericide is introduced, we need to find the derivative N'(t) and evaluate it at t = 3. Taking the derivative, we have:
N'(t) = -9400(2t)/(1 + t²)².
Evaluating N'(t) at t = 3, we get:
N'(3) = -9400(2(3))/(1 + 3²)² = -9400(6)/(1 + 9)² = -9400(6)/10² = -9400(6)/100 = -5640.
Therefore, the rate of change of the number of bacteria in the culture 3 minutes after the bactericide is introduced is -5640 bacteria/min.
(b) To find the population of the bacteria in the culture 3 minutes after the bactericide is introduced, we plug in t = 3 into the function N(t):
N(3) = 9400/(1 + 3²) = 9400/(1 + 9) = 9400/10 = 940.
Therefore, the population of the bacteria in the culture 3 minutes after the bactericide is introduced is 940 bacteria.
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Related Questions
write the equation of the line in fully simplified slope-intercept form
Answers
y= -4x-5 you can find this by seeing how far it goes up/down and where it starts:)
Jane, kevin, and hans have a total of in their wallets. kevin has less than jane. hans has times what jane has. how much does each have?
Answers
Based on the given conditions, Jane has $31, Kevin has $25, and Hans has $50 in their wallets.
Let's solve the problem step by step.
First, let's assume that Jane has X dollars in her wallet. Since Kevin has $6 less than Jane, Kevin would have X - $6 dollars in his wallet.
Next, we're given that Hans has 2 times what Kevin has. So, Hans would have 2 * (X - $6) dollars in his wallet.
According to the information given, the total amount of money they have in their wallets is $106. We can write this as an equation:
X + (X - $6) + 2 * (X - $6) = $106
Simplifying the equation:
4X - $18 = $106
4X = $124
X = $31
Now we know that Jane has $31 in her wallet.
Substituting this value into the previous calculations, we find that Kevin has $31 - $6 = $25 and Hans has 2 * ($25) = $50.
To find the total amount they have, we sum up their individual amounts:
Jane: $31
Kevin: $25
Hans: $50
Adding these amounts together, we get $31 + $25 + $50 = $106, which matches the total amount stated in the problem.
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The complete question is:
Jane, kevin and hans have a total of $106 in their wallets. kevin has $6 less than Jane. hans has 2 times what kevin has. how much do they have in their wallets?
At this point, let us extend what you have done in Optimize. Consider the sets of
geometric figures below then answer the questions that follow.
10
1.
2.
3.
4.
5.
1
m
Q
How many points do OO and line / have in common?
How many points do OP and line m have in common?
How many points do OQ and line n have in common?
What kind of angle is formed by the line segments from the center of the
circles and their corresponding lines?
Are the line segments from the center of the circles parallel to their respective
lines? If not, write the correct term.
Answers
Answer:
No point,One Point.Reflex Angle Formed .
Use an equation to solve each percent problem. Round your answer to the nearest tenth, if necessary.
80% of 58 is what?
Answers
Compounds A and B are used in an experiment.
The equation y=500(0.5)x represents the remaining amount of compound A, in grams, after x hours. The table shows the
remaining amount of compound B.
What is the positive difference in the initial amount of each compound?
Answers
The positive difference in the initial amount of each compound is 100
The positive difference in the initial amount of each compoundFrom the question, we have the following parameters that can be used in our computation:
y = 500(0.5)ˣ
The initial amount is when x = 0
So, we have
y = 500(0.5)⁰
y = 500
From the table of values, we have
Initial amount of B = 200/0.5
Initial amount of B = 400
So, the difference is
Difference = 500 - 400
Difference = 100
Hence, the positive difference is 100
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Question
Compounds A and B are used in an experiment.The equation y=500(0.5)x represents the remaining amount of compound A, in grams, after x hours. The table shows the remaining amount of compound B.
Time (h) Grams of B
1 200
2 100
3 50
What is the positive difference in the initial amount of each compound?
2. Consider the system defined by the impulse response h(n)=28(n+3)+28(n)+28(n-3). a) b) c) d) z Represent h(n). (1 v.) Characterize the system in terms of causality and stability. Justify. (1 v.) Determine the frequency response of the system H(ew). (1 v.) Represent module and phase of the system. (1 v.)
Answers
The system defined by the impulse response h(n) = 28(n+3) + 28n + 28(n-3) can be represented as h(n) = 28δ(n+3) + 28δ(n) + 28δ(n-3), where δ(n) denotes the unit impulse function.
In terms of causality, we can determine whether the system is causal by examining the impulse response. If the impulse response h(n) is non-zero only for n ≥ 0, then the system is causal. In this case, since the impulse response h(n) is non-zero for n = -3, 0, and 3, the system is not causal.
To determine the stability of the system, we need to examine the summation of the absolute values of the impulse response. If the summation is finite, the system is stable. In this case, we can calculate the summation as ∑|h(n)| = 28 + 28 + 28 = 84, which is finite. Therefore, the system is stable.
However, since the impulse response is given in the time domain and not in a closed-form expression, it is not possible to directly determine the frequency response without further manipulation or additional information.
Given the absence of specific frequency domain information or a closed-form expression for the frequency response, it is not possible to accurately represent the module and phase of the system H(e^ω) without further calculations or additional details about the system.
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how many acres is a lot that totals 19,602 square feet...?
Answers
The area covered by 19, 602 square feet in acres is around 0.45, according to the relation between the two.
As per the known relation between acre and square foot, the two quantities are related through the formula -
1 acre is equal to 43, 560 square feet
Therefore, formula to convert square feet to acre will be -
Amount of area in acres = area covered in square feet/unit equivalent area in square feet
So, rewriting the above mentioned relation.
43, 560 square feet is equal to 1 acre
Thus, 19, 602 square feet will be equal to acres = 19, 602/43, 560
Performing division on Right Hand Side of the equation
The amount of area in acres = 0.45 acres
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Solve linear equation
- 0.75p - 2 =0.25p
Answers
Answer: p = -2
Step-by-step explanation:
- 0.75p - 2 =0.25p add .75p on both sides
-2 = p That's the same thing at p = -2
Answer:
p = -2
Step-by-step explanation:
-0.75p - 2 = 0.25p Subtract 0.25p from both sides
-1p - 2 = 0 Add 2 to both sides
-1p = 2 Divide by -1 on both sides
p = -2
How many solutions exist for this system of equations y=x-3 y=x^2-6x+10
A. 0
B. 1
C. 2
D. Infinite
Answers
Has No Solution
Answer:
0
Step-by-step explanation:
Select the correct answer from each drop-down menu. What is the distance and midpoint between points F and G? A number line ranges from negative 4 to 6. Points F and G are plotted on the number line at minus 3 and 1.6. distance = midpoint =
Answers
The distance between F and G is 4.6
The midpoint between F and G is -0.7.
What is a number line?It is the representation of numbers in real order.
The difference between the consecutive numbers in a number line is always positive.
We have,
Point F is at -3.
Point G is at 1.6.
Now,
The distance between F and G.
= 1.6 - (-3)
= 1.6 + 3
= 4.6
Now,
The midpoint between F and G.
There are 4.6 points between F and G.
So,
4.6/2 = 2.3
Midpoint.
=-3 + 2.3
= -0.7
Thus,
The distance is 4.6
The midpoint is -0.7.
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How many square inches of cloth are cut from the square (n = 3.14) if it’s 38inches
Answers
The number of square inches of cloth cut from the square will be 1,017.36 square inches. Then the correct option is A.
What is the area of the circle?It is the close curve of an equidistant point drawn from the center. The radius of a circle is the distance between the center and the circumference.
Let r be the radius of the circle. Then the area of the circle will be
A = πr² square units
The radius is given as,
r = 36 / 2
r = 18 inches
The area of the circle is given as,
A = π x (18)²
A = 3.14 x 324
A = 1,017.36 square inches
The number of square inches of cloth cut from the square will be 1,017.36 square inches. Then the correct option is A.
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The complete question is given below.
A circle is cut from a square piece of cloth, as shown:
A square, one side labeled as 36 inches, has a circle inside it. The circle touches all the sides of the square. The portion of the square outside the circle is shaded.
How many square inches of cloth is cut from the square?
(π = 3.14)
1,017.36 in2
1,489.24 in2
1,182.96 in2
1,276.00 in2
three different are rung at intervals of 36 minutes 40 minutes and 48 minutes if they were rung together at 9 a.m. at what time would they be rung together again?
Answers
To find the time in which all the bells ring together, we need to find the LCM of 36,40,48.
Prime factorization of 36=2×2×3×3
Prime factorization of 40=2×2×2×5
Prime factorization of 48=2×2×2×2×3
Hence, LCM of 36,40,48=2×2×2×2×3×3×5=720 seconds.
720seconds=
60
720
minutes=12minutes
Hence, all the bells will ring together after 12 mi
Answer:
9pm or 12 hours from 9am.
Find the root of the equation -x+ sin(x) cos (x) = 0 using bisection algorithm. Perform two iterations using starting interval a = 0, b = 1. Estimate the error.
Answers
The bisection algorithm, the root of the equation -x + sin(x) cos(x) = 0 is approximately 0.841523 ± 0.03125.
Given equation:
-x + sin(x) cos(x) = 0
Using the bisection algorithm,
Let the function be f(x) = -x + sin(x) cos(x)
An interval [a, b] = [0, 1]
Therefore, the value of the function at a :
f(a) = -0 + sin(0) cos(0)
= 0 and at b is
f(b) = -1 + sin(1) cos(1)
≈ -0.17
The root of the function is between the interval [a, b].
The root of the equation is the point x such that f(x) = 0.
Estimating the error:
After the nth iteration, the error is given by:
|E_n| ≤ |b_n - a_n| / 2^(n+1), where b_n and a_n are the endpoints of the interval at the nth iteration.
Now, using the bisection algorithm, the two iterations using the starting interval a = 0, b = 1 are performed.
= |E_2| ≤ |b_2 - a_2| / 2^(2+1)
= |0.25| / 2^3
= 0.03125
The bisection method is relatively slow but robust and can solve a wide range of nonlinear equations. Therefore, using the bisection algorithm, the root of the equation -x + sin(x) cos(x) = 0 is approximately 0.841523 ± 0.03125.
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what is the answer for this question
5 days 18hours=
Answers
Answer: he only way it answers is how many days, hours, and minutes. It would be nice if it also told me the total amount of hours also. Ahil 2021-09-30 01:09:28. Can you add a days section
Step-by-step explanation:
Which value is the solution to the equation - 8 = 6-7x?
A. -7
B. - 2
c. 2
D. 5
Answers
Answer:
-8=6-7x (subtract 6 from both sides)
-14=-7x (divide each side by -7)
x=2
the answer is C
Answer:
c: 2
Step-by-step explanation:
-8 = 6 -7x, isolate the -7 by moving the 6 to the left side of the equation so it becomes -8-6 = -7x --> -14 = -7x and then divide by both sides by -7 and then you get x =2, you can plug in the number back into the equation to make sure
Find the area under the standard normal curve to the left of z=−1.5 z = − 1.5 and to the right of z=−1.1 z = − 1.1 . Round your answer to four decimal places, if necessary.
Answers
The task is to find the area under the standard normal curve to the left of z = -1.5 and to the right of z = -1.1, rounded to four decimal places.
The area under the standard normal curve represents the probability of a random variable being less than or greater than a certain value. To find the area to the left of z = -1.5, we can look up the corresponding cumulative probability in the standard normal distribution table or use statistical software.
Similarly, to find the area to the right of z = -1.1, we can calculate 1 minus the cumulative probability to the left of -1.1. By subtracting the area to the right from the area to the left, we can determine the desired area under the standard normal curve.
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use the ratio test to determine whether the series is convergent or divergent. [infinity] n 7n n = 1 identify an.
Answers
To determine whether the series ∑(n=1 to infinity) 7n/n is convergent or divergent, we can apply the ratio test. The ratio test helps us determine the convergence or divergence of a series by examining the limit of the ratio of consecutive terms.
In this case, let's calculate the ratio of consecutive terms using the formula for the ratio test:
lim(n→∞) |(7(n+1)/(n+1))/((7n/n)|
Simplifying the expression, we get:
lim(n→∞) |7(n+1)/n|
As n approaches infinity, the limit evaluates to:
lim(n→∞) |7(n+1)/n| = 7
Since the limit is a finite positive value (7), which is less than 1, the ratio test tells us that the series is convergent.
However, you mentioned identifying an (term) in the series, and it seems there may be an incomplete part of the question. Please provide additional information or clarification about identifying an term in the series so that I can provide a more specific answer.
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Find the length of LM.
Write your answer as a whole number or a decimal
Answers
Answer:
LM = 5
Step-by-step explanation:
A secant is a straight line that intersects a circle at two points.
A segment is part of a line that connects two points.
Intersecting Secants TheoremThe product of the measures of one secant segment and its external part is equal to the product of the measures of the other secant segment and its external part.
The given diagram shows two secant segments LN and PN that intersect at exterior point N. Their external parts are MN and ON, respectively.
Therefore, according to the Intersecting Secants Theorem:
\(\implies \sf LN \cdot MN =PN \cdot ON\)
Given values:
LN = 3 + x - 2 = x + 1MN = 3PN = 4 + x - 5 = x - 1ON = 4Substitute the values into the equation and solve for x:
\(\implies \sf LN \cdot MN =PN \cdot ON\)
\(\implies (x+1)\cdot 3=(x-1)\cdot 4\)
\(\implies 3x+3=4x-4\)
\(\implies 4x-4=3x+3\)
\(\implies 4x-4-3x=3x+3-3x\)
\(\implies x-4=3\)
\(\implies x-4+4=3+4\)
\(\implies x=7\)
To determine the length of LM, substitute the found value of x into the expression for the line segment:
\(\implies \overline{LM}=7-2=5\)
Therefore, the length of LM is 5.
Geometric number between 12 and 8.
Answers
Answer:
9.79
Step-by-step explanation:
12 × 8 and root of that is 9.7979589
(a) Latoya lost 60 dollars from her pocket.
Write a signed number to represent this change.
(b) A car corporation produced 1180 more cars this month than last.
Write a signed number to represent this month's change in production.
Answers
Answer:
-60, +1180
Step-by-step explanation:
7.45 x 103 = ___
7.45
74.5
7,450
74,500
Answers
Answer: 745
Step-by-step explanation:
\(7.45 \times 10^{3}=7.45(1000)=\boxed{745}\)
Given f(x) = 70 + 11x, find x when f(x) =158
Answers
Answer:
\( \huge \boxed{x = 8}\)
Step-by-step explanation:
f(x) = 70 + 11x
To find the value of x when f(x) =158 , equate f(x) to 158
We have
\(70 + 11x = 158 \\ 11x = 158 - 70 \\ 11x = 88\)
Divide both sides by 11
\( \frac{11x}{11} = \frac{88}{11} \\ \)
We have the final answer as
x = 8Hope this helps you
Under what circumstances should you look at pairs of negative factors of the constant term when factoring a trinomial of the form x² + bx+c?
Answers
The circumstances are for when the leading coefficient and the constant term are positive
How to determine the circumstancesUnder specific conditions, it is possible to examine pairs of negative factors related to the constant term (c) when factoring a trinomial in the format of x² + bx + c.
When the constant term (c) and the leading coefficient (the coefficient of x²) both have positive values, it is possible to factor the trinomial as a product of two binomials that have negative factors.
The reason for this stems from the fact that when the negative factors are added together, the result is the negative coefficient (b) of the middle term.
To factorize a trinomial, one can identify the negative factors that pair up to equal the constant term. This will enable the expression to be factored as (x - p)(x - q) with p and q being those negative factors.
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A researcher plans to use a random sample of houses to estimate the mean size (in square feet) of the houses in a large population. The researcher is deciding between a 95% confidence level and a 99% confidence level. Compared with a 95% confidence interval, a 99% confidence interval will be
Answers
Considering the margin of error, it is found that the 99% confidence interval will be wider than the 95% confidence interval.
What is the margin of error of a confidence interval?It is given by:
\(M = z\frac{s}{\sqrt{n}}\)
In which:
z is the critical value, which increases with a higher confidence level.s is the standard deviation.n is the sample size.From this, we get that the margin of error is direct proportional to the critical value, hence \(z_{99}>z_{95}\), which means that the 99% confidence interval will be wider, that is, it will have a larger margin of error, than the 95% confidence interval.
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How many square centimeters are in an area of 11.6inch
2
? Express your answer numerically in square centimeters.
Answers
The area of 11.6 square inches is equivalent to approximately 74.84 square centimeters.
To convert square inches to square centimeters, we need to use the conversion factor 1 inch = 2.54 centimeters. Since we are dealing with areas, we need to square this conversion factor as well.
Convert inches to centimeters:
11.6 inches * 2.54 centimeters/inch = 29.464 centimeters
Square the conversion factor:
(2.54 centimeters/inch) * (2.54 centimeters/inch) = 6.4516 square centimeters/square inch
Calculate the area in square centimeters:
29.464 centimeters * 6.4516 square centimeters/square inch = 74.84 square centimeters.
Rounding this value to two decimal places, the area of 11.6 square inches is approximately 190.29 square centimeters.
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find all abelian groups (up to isomorphism) of order 360
Answers
The abelian groups (up to isomorphism) of order 360 are:
1. C360: The cyclic group of order 360.2. C2 x C2 x C3 x C3 x C5: The direct product of cyclic groups of orders 2, 2, 3, 3, and 5.
How can we determine the abelian groups of order 360?To find all abelian groups of order 360, we need to consider the prime factorization of 360, which is 2^3 × 3^2 × 5. The abelian groups will be direct products of cyclic groups whose orders divide 360.
First, we consider the cyclic groups. Since the order of an element in a cyclic group divides the order of the group, we can have cyclic subgroups of orders 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360.
Now, we can combine these cyclic subgroups to form the abelian groups. We can take direct products of the cyclic groups to obtain different possibilities. The order of the resulting group will be the product of the orders of the cyclic subgroups.
In this case, we can form the following abelian groups of order 360:
C360: This is the cyclic group of order 360 itself.
C2 x C2 x C3 x C3 x C5: This is the direct product of cyclic groups of orders 2, 2, 3, 3, and 5.
These are the two abelian groups (up to isomorphism) that can be formed using the prime factorization of 360.
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What is the value of the expression (5-2 √3)(5+2 √{3) ?
Answers
The value of the expression (5 - 2√3)(5 + 2√3) is 13. To find the value of the expression (5 - 2√3)(5 + 2√3), we can use the difference of squares formula.
The difference of squares formula states that for any numbers a and b, (a - b)(a + b) is equal to \(a^2 - b^2.\)
In this case, let's let a = 5 and b = 2√3.
Now, we can substitute these values into the formula:
\((5 - 2√3)(5 + 2√3) = 5^2 - (2√3)^2\)
Simplifying further:
= 25 - 4(3)
= 25 - 12
= 13
Therefore, the value of the expression (5 - 2√3)(5 + 2√3) is 13.
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Gavin can type 3600 words in 60 minutes.
Dan can type 3850 in 70 minutes.
Colin can type 3500 words in 70 minutes.
Who types at the fastest rate of words per minute?
Answers
Answer:
Gavin
Step-by-step explanation:
Gavin: 3600 words / 60 minutes = 60 wpm
Dan: 3850 words / 70 minutes = 55 wpm
Colin: 3500 words / 70 minutes = 50 wpm
i need help please thanks
Answers
Answer:
So I think the answer for the first one is A and the second one is D
Step-by-step explanation:
I hope those help
The Dow Jones Industrial Average (DJIA) represents an average stock price of many corporations
Suppose the DJIA was 34, 117 points at 9: 00 am but was losing 0.7% of its value every minute.
At that rate, what would be the DJIA value at 9: 10 am?
Answers
The DJIA represents an average stock price value at 9:10 am would be approximately 31,802.65 points.
What is average?
The average, also known as the arithmetic mean, is a measure of central tendency that represents the typical value in a set of numbers. It is calculated by adding up all the values in the set and then dividing by the total number of values.
If the DJIA is decreasing by 0.7% of its value every minute, this means that it is losing 0.007 times its value every minute.
To determine the DJIA value at 9:10 am, we need to calculate the number of minutes that have elapsed since 9:00 am.
There are 10 minutes from 9:00 am to 9:10 am, so the DJIA value at 9:10 am can be calculated as follows:
DJIA value at 9:10 am = 34,117 x \((1 - 0.007)^{10\)
= 34,117 x \((0.993)^{10\)
= 34,117 x 0.933
= 31,802.65
Therefore, the DJIA value at 9:10 am would be approximately 31,802.65 points.
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