Write the first and second partial derivatives.
g(r,t)=t ln r + 12 rt^7 − 3(9^r)
g_r = ______
g_rr = _______
g_rt = ______
g_t = _____
g_tr = _____
g_tt = ______
Answers
To find the first and second partial derivatives of the function \(g(r,t) = t \ln r + 12rt^7 - 3(9^r)\), we differentiate with respect to each variable.
First partial derivatives:
\(g_r = \frac{\partial g}{\partial r} = \frac{\partial}{\partial r}(t \ln r + 12rt^7 - 3(9^r))\)
Differentiating each term separately:
\(g_r = \frac{\partial}{\partial r}(t \ln r) + \frac{\partial}{\partial r}(12rt^7) - \frac{\partial}{\partial r}(3(9^r))\)
Using the derivative rules:
\(g_r = t \cdot \frac{1}{r} + 12t^7 - 3 \cdot (\ln 9) \cdot (9^r) \cdot (\ln 9^r)\)
Simplifying:
\(g_r = \frac{t}{r} + 12t^7 - 3(\ln 9)(9^r)(r \ln 9)\)
Second partial derivatives:
\(g_{rr} = \frac{\partial^2 g}{\partial r^2} = \frac{\partial}{\partial r}\left(\frac{t}{r} + 12t^7 - 3(\ln 9)(9^r)(r \ln 9)\right)\)
Differentiating each term separately:
\(g_{rr} = \frac{\partial}{\partial r}\left(\frac{t}{r}\right) + \frac{\partial}{\partial r}\left(12t^7\right) - \frac{\partial}{\partial r}\left(3(\ln 9)(9^r)(r \ln 9)\right)\)
Using the derivative rules:
\(g_{rr} = -\frac{t}{r^2} + 0 - 3(\ln 9)(9^r)\left(\ln 9 + r \cdot \frac{1}{9} \cdot 9^{-r}\right)\)
Simplifying:
\(g_{rr} = -\frac{t}{r^2} - 3(\ln 9)(9^r)\left(\ln 9 + \frac{r}{9} \cdot 9^{-r}\right)\)
\(g_{rt} = \frac{\partial^2 g}{\partial r \partial t} = \frac{\partial}{\partial r}\left(\frac{\partial g}{\partial t}\right)\)
Differentiating the first partial derivative \(g_r\) with respect to \(t\):
\(g_{rt} = \frac{\partial}{\partial t}\left(\frac{t}{r} + 12t^7 - 3(\ln 9)(9^r)(r \ln 9)\right)\)
\(g_{rt} = \frac{1}{r} + 84t^6 - 3(\ln 9)(9^r)(\ln 9)\)
\(g_t = \frac{\partial g}{\partial t} = \frac{\partial}{\partial t}(t \ln r + 12rt^7 - 3(9^r))\)
Differentiating each term separately:
\(g_t = \frac{\partial}{\partial t}(t \ln r) + \frac{\partial}{\partial t}(12rt^7) - \frac{\partial}{\partial t}(3(9^r))\)
Using the derivative rules:
\(g_t = \ln r + 12r(7t^6) + 0\)
Simplifying:
\(g_t = \ln r + 84rt^6\)
\(g_{tr} = \frac{\partial^2 g}{\partial t \partial r} = \frac{\partial}{\partial t}\left(\frac{\partial g}{\partial r}\right)\)
Differentiating the first partial derivative \(g_r\) with respect to \(t\):
\(g_{tr} = \frac{\partial}{\partial t}\left(\frac{t}{r} + 12t^7 - 3(\ln 9)(9^r)(r \ln 9)\right)\)
\(g_{tr} = 0 + 84r(6t^5) - 3(\ln 9)(9^r)(\ln 9)(r \ln 9)\)
\(g_{tt} = \frac{\partial^2 g}{\partial t^2} = \frac{\partial}{\partial t}\left(\ln r + 84rt^6\right)\)
\(g_{tt} = 0 + 84r(6)(t^5)\)
Simplifying:
\(g_{tt} = 504rt^5\)
Therefore, the first and second partial derivatives of \(g(r,t) = t \ln r + 12rt^7 - 3(9^r)\) are:
\(g_r = \frac{t}{r} + 12t^7 - 3(\ln 9)(9^r)(r \ln 9)\)
\(g_{rr} = -\frac{t}{r^2} - 3(\ln 9)(9^r)\left(\ln 9 + \frac{r}{9} \cdot 9^{-r}\right)\)
\(g_{rt} = \frac{1}{r} + 84t^6 - 3(\ln 9)(9^r)(\ln 9)\)
\(g_t = \ln r + 84rt^6\)
\(g_{tr} = 84r(6t^5) - 3(\ln 9)(9^r)(\ln 9)(r \ln 9)\)
\(g_{tt} = 504rt^5\)
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Related Questions
pls answer if you can i need the answer right away
Answers
Answer:
What’s the question?
Step-by-step explanation:
if the output and input of a linear equation are proportional, whare will the graph of the equation cross the x axis
Answers
Answer:
at x=0 (the origin)
Step-by-step explanation:
The equation for a proportional relationship is ...
y = kx
The equation of a straight line is ...
y = kx +b . . . . . for some y-intercept b
Comparison to general lineComparing the proportional relation to the relation for a general line, we see that b=0 in the proportional relation. That is, the y-intercept is at y=0, the origin of the Cartesian coordinate plane.
The graph crosses the x-axis at (0, 0), or x=0.
__
Additional comment
In y=kx, the value k is called "the constant of proportionality." It is also the slope of the line on a graph.
In the usual representation of the slope-intercept equation of a line, y=mx+b, the slope of the line is represented by 'm'. In the above, we used 'k' for that purpose, to facilitate comparing the equations.
The Parent-Teacher Association (PTA) wants to find out
how much students pay for school supplies at the
beginning of the school year, so it conducts a survey at
certain local stores. PTA members visit these stores to
find the prices of the items on the students' supply lists.
What is the sample in this study?
A. The prices of listed items at the selected stores
B. The prices of pencils at the selected stores
C. The prices of listed items at all local stores
D. The prices of listed items at one store
Answers
Write an equation for this graph in slope-intercept
(Please show proof on how u did it i need to show work)
thank u so much (:
Answers
to get the equation of any straight line, we simply need two points off of it, let's use those two in the picture below.
\((\stackrel{x_1}{3}~,~\stackrel{y_1}{7})\qquad (\stackrel{x_2}{6}~,~\stackrel{y_2}{8}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{8}-\stackrel{y1}{7}}}{\underset{\textit{\large run}} {\underset{x_2}{6}-\underset{x_1}{3}}} \implies \cfrac{ 1 }{ 3 }\)
\(\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{7}=\stackrel{m}{ \cfrac{ 1 }{ 3 }}(x-\stackrel{x_1}{3}) \\\\\\ y-7=\cfrac{ 1 }{ 3 }x-1\implies {\Large \begin{array}{llll} y=\cfrac{ 1 }{ 3 }x+6 \end{array}}\)
9. A random variable X is distributed according to X~ N(= 25,0² =9) (a) Determine such M so that P(X < M) = 0.95. (b) Determine the median.
Answers
The standard normal distribution has a mean of 0 and a standard deviation of 1. M ≈ 30.935. The median of the distribution is also 25.
(a) To find M, we first need to convert the given values of mean and standard deviation to the standard normal distribution. This can be done by using the formula Z = (X - μ) / σ, where Z is the Z-score, X is the value of interest, μ is the mean, and σ is the standard deviation. In this case, we have X ~ N(25, 9). Substituting the values into the formula, we get Z = (X - 25) / 3. Now we need to find the Z-score that corresponds to the desired probability of 0.95. Using a standard normal distribution table or a calculator, we find that the Z-score corresponding to a cumulative probability of 0.95 is approximately 1.645. Setting Z equal to 1.645, we can solve for X: (X - 25) / 3 = 1.645. Solving for X, we get X ≈ 30.935. Therefore, M ≈ 30.935.
(b) The median is the value that divides the distribution into two equal halves. In a normal distribution, the median is equal to the mean. In this case, the mean is given as 25. Therefore, the median of the distribution is also 25.
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Please help and show work, will give lots of points!
Lourdes is reading a biography for her history class. She reads 30 pages each day. After 9 days, Lourdes has read 3/5 of the biography. Write a linear equation to represent the number of pages Lourdes still has to read after x days.
y = []x + []
(Use above format to write the equation.)
What does the y-intercept of this linear equation represent?
A. Pages already read
B. Pages in book
C. Pages read each day
D. Days to finish
Answers
Answer:
The linear equation is y = 450 - 30 x, where y is the number of pages
Lourdes has left to read after x days
Step-by-step explanation:
Each day, Lourdes reads 30 pages of a 450-page book
- We need to write a linear equation to represent the number of pages
Lourdes has left to read after x days
∵ Lourdes reads 30 pages each day
∵ Lourdes will read for x days
∴ The number of pages Lourdes will read in x day = 30 x
- The left pages will be the difference between the total pages of the
book and the pages Lourdes read
∵ The book has 450 pages
∵ Loured will read 30 x in x days
∴ The number of pages left = 450 - 30 x
- Assume that y represents the number of pages Lourdes has left
to read after x days
∴ y = 450 - 30 x
The linear equation is y = 450 - 30 x, where y is the number of
pages Lourdes has left to read after x days
there is 25 people on a bus they pay R x each if there where 5 more people every body can pay R3 less how much is R x
Answers
Based on the given parameters, the value of Rx is R18
How to determine the value of Rx?The given parameters are:
Number of people = 25
Amount paid = Rx
Additional people of 5 = Amount paid being R3 less
The total amount paid is calculated using:
Total amount = Number of people * Amount paid
Substitute the known values in the above equation
Total amount = 25 * x
Total amount = (25 + 5) * (x - 3)
Substitute Total amount = (25 + 5) * (x - 3) in Total amount = 25 * x
(25 + 5) * (x - 3) = 25 * x
Evaluate the sum
30 * (x - 3) = 25 * x
Evaluate the products
30x - 90 = 25x
Collect like terms
30x - 25x = 90
Evaluate the like terms
5x = 90
Divide both sides by 5
x = 18
Hence, the value of Rx is R18
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Which two numbers doesstartroot 128 endroot lie between on a number line?.
Answers
Therefore, √128 lies between 11 and 12 on a number line.
To determine the two numbers between which √128 lies on a number line, we can calculate the square roots of consecutive perfect squares that surround 128.
By calculating the square roots, we can find the two nearest whole numbers that √128 lies between.
Calculating the square roots of perfect squares near 128:
√121 = 11
√144 = 12
Since 128 is greater than 121 and less than 144, √128 lies between √121 and √144 on the number line.
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i need help if yall can help me
Answers
Answer:
C. 5.8
Step-by-step explanation:
We use the Pythagoras theorem here. which is a²+b²=c².
a and b are the replacements for the 5miles and 3miles here.
They are a and b because a and b are always on the right angle side.
The longest side of this triangle is called a hypotenuse which is the c in this case.
the other sides (a and b) are called the adjacent and the opposite.
it doesn't matter if you use adjacent or opposite for a and b in this case.
the only important thing is the hypotenuse here.
So it's a²+b²=c², which is 5²+3²=c².
5² is 25 and 3² is 9, so we add them and we get 34.
so c²=34, we don't want the square on c so we send it to 34 which becomes √34.
√34 gives us 5.8 miles which is up to just only one decimal place.
You might need to use calculators for square roots because they take long to calculate.
On a biased dice the probability of getting a 6 is 4/5 the dice is rolled 500 times how many sixes would you expect to roll
Answers
Answer:
400 sixes
Step-by-step explanation:
Find how many sixes you could expect to roll by multiplying 500 by 4/5:
500(4/5)
= 400
So, you would expect to roll 400 sixes
What is the inverse of f(x)=3x^2
Answers
Answer:
f^-1 (x)=sqrt(x/3), -sqrt(x/3).
Step-by-step explanation:
f(x)=3x^2
y=3x^2
x=3y^2
y^2=x/3
y=sqrt(x/3), -sqrt(x/3).
Answer:
\(y=3x^2\)
\(y=3x^2\)
Step-by-step explanation:
There is no step-by-step explanation because it's a graph. Well, it's how I did it. Therefore, the graph in the picture that I'm about to post will be your answer.
Shawna joined the bowling league at Bronze Lanes, a bowling alley. She scores a strike 75% of the time. For a perfect game, she needs to bowl a strike 12 times in a row. How likely is it that Shawna will bowl a perfect game at her next game?
Shawna simulates the situation by using a computer to randomly generate 12 numbers from 1 to 4. Each time 1, 2, or 3 appears, it represents a strike
Answers
The probability of Shawna bowling a perfect game is 7.23%.
Assuming that Shawna's chance of getting a strike is independent on each roll (which may not be entirely accurate in real life), the probability of her getting a strike on any given roll is 0.75.
To calculate the probability of Shawna bowling a perfect game (i.e., getting a strike on all 12 rolls), we can use the multiplication rule of probability, which states that the probability of two independent events occurring together is the product of their individual probabilities.
So, the probability of Shawna getting a strike on the first roll is 0.75. The probability of her getting a strike on the second roll, assuming she got a strike on the first roll, is also 0.75. Similarly, the probability of her getting a strike on the third roll, assuming she got strikes on the first two rolls, is also 0.75. We can continue this pattern for all 12 rolls.
Therefore, the probability of Shawna bowling a perfect game is:
0.75 x 0.75 x 0.75 x ... (12 times)
which is approximately equal to 0.0723, or about 7.23%.
Using a computer to randomly generate 12 numbers from 1 to 4 and treating 1, 2, or 3 as a strike is a reasonable way to simulate the situation and estimate the probability of Shawna bowling a perfect game. By repeating this process many times, we can get a better idea of the distribution of possible outcomes and the likelihood of different results.
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Six times a number is at least four less than eight times the number. If x represents the number, which will help find
the number?
6x < 8x-4
6x8x-4
6x8x-4
O 6x> 8x-4
Answers
Answer:
6x ≥ 8x - 4
Step-by-step explanation:
I need help with this
Answers
Answer:
8)2.86 cm^2
9) 100.48 m^2
10) 23.08 in^2
Step-by-step explanation:
8) In the figure , there's a right angled triangle in which a circle of radius 1 cm is inscribed in that triangle.
Area of the right angled triangle =
\( \frac{1}{2} \times 4 \times 3 = 6 \: {cm}^{2} \)
Area of the circle =
\(\pi {r}^{2} = \pi {(1)}^{2} = \frac{22}{7} = 3.14 \: {cm}^{2}
\)
Area of the shaded portion ( although itz not shaded..... i mean remaining portion ) = 6 - 3.14 =2.86cm^2
9) In this figure there's a circle in which more 2 circles are inscribed in such a way that the center of the bigger circle is in the circumferences of both the circles.
Area of the bigger circle =
\(\frac{\pi {d}^{2} }{4} = \frac{\pi \times 16 \times 16}{4} = 64\pi \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 200.96 {m}^{2} \)
The small circles have equal diameter ( i.e. 8m) . So , Area of the small circles =
\(2( \frac{\pi {d}^{2} }{4} ) = 2( \frac{\pi \times {8}^{2} }{4} ) = 32\pi \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 100.48 \: {m}^{2} \)
Area of the shaded region =
200.96 - 100.48 = 100.48 m^2
10) In the figure , the radius of the circle is 9 in.
So the area of a quarter =
\( \frac{\pi {r}^{2} }{4} = \frac{\pi \times {(9)}^{2} }{4} = \frac{81\pi}{4} = 63.58 \: {in}^{2} \)
Also in that circle a right angled triangle is formed. Area of that right angles triangle =
\( \frac{1}{2} \times 9 \times 9 = 40.5 \: {in}^{2} \)
So the area of the region shaded =
63.58 - 40.5 = 23.08 in^2
In △OAB , points C and E lie on OA¯¯¯¯¯¯¯¯ , points D and F lie on OB¯¯¯¯¯¯¯¯ , and CD¯¯¯¯¯¯¯¯∥EF¯¯¯¯¯¯¯¯∥AB¯¯¯¯¯¯¯¯ . If OC=3 , EA=5 , OD=6 , and DB=14 , what is CE ?
Answers
Answer:
2
Step-by-step explanation:
if 26 children were to be born in a hospital on a given day, how many combinations of 6 boys and 20 girls would exist? 230,230 4 x 10^26 500,000 15 Z
Answers
The number of combinations of 6 boys and 20 girls that can exist among 26 children born in a hospital on a given day is 230,230.
]To calculate the number of combinations, we can use the concept of binomial coefficients. The formula for calculating the number of combinations is C(n, k) = n! / (k!(n-k)!), where n is the total number of objects and k is the number of objects we want to select.
In this case, we have 26 children in total, and we want to select 6 boys and 20 girls. Plugging these values into the formula, we get C(26, 6) = 26! / (6!(26-6)!) = 230,230. Therefore, there are 230,230 different combinations of 6 boys and 20 girls that can exist among the 26 children born in the hospital on that given day.
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Which of the following answers is correct?
Answers
The function d/dx (f/g) = 2 gives the value of c as -4.
What is Quotient Rule?The denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator, is how the Quotient Rule defines the derivative of a quotient.
Given:
f(x) = 1/2x and g(x) = 1/ (3x + c)
Using Quotient rule
d/ dx (f/g)
= g f' - f g- / g²
So, d/dx ( 3x+c / 2x )
= [3x -1(3x+c)] / 2x²
= -c/ 2x²
Now out x= 1
Also, d/dx (f/g)= 2
-c/ 2 = 2
c= -4.
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Lin solved the equation 8(x-3)+7=2x(4-17) incorrectly.
Find the errors in her solution. What should her answer have been?
Lin's Steps - 8 (x-3) + 7 = 2x (4 - 17)
8 (x-3) + 7 = 2x (13)
8x - 24 + 7 = 26x
8x - 17 = 26x
-17 = 34x
-1/2 = x
Answers
Answer:
22
Step-by-step explanation:
Answer:
1/2 = x
Step-by-step explanation:
8 (x-3) + 7 = 2x (4 - 17)
8 (x-3) + 7 = 2x (-13)
8x - 24 + 7 = -26x
8x - 17 = -26x
-17 = -34x
1/2 = x
A teacher was comparing two sets of quiz scores shown below. What reasonable conclusion can be drawn from the data?
a. The variability of Quiz 2 scores is greater than the variability of Quiz 1 scores, but the mean is the same.
b. The variability of Quiz 1 scores is greater than the variability of Quiz 2 scores, but the mean is the same.
c. The mean of Quiz 1 scores is greater than the mean of Quiz 2 scores, but the variability is the same.
d. The mean of Quiz 2 scores is greater than the mean of Quiz 1 scores, but the variability is the same.
Answers
Based on the given data, it can be concluded that the mean of Quiz 1 scores is greater than the mean of Quiz 2 scores, but the variability is the same.
This can be inferred from the fact that the mean score for Quiz 1 is 80, which is higher than the mean score for Quiz 2, which is 75. However, the range of Quiz 1 scores is 10 points, while the range of Quiz 2 scores is 5 points, indicating that the variability of Quiz 1 scores is greater than the variability of Quiz 2 scores.
Therefore, the correct option is (c).
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Shade one more square to make a pattern with 1 line of symmetry
Answers
Answer:
Shade to form a U shape
Step-by-step explanation:
U shape has only 1 line of symmetry
HELP MEE IN MATHHHH pleasee
Answers
Answer:
44.7
Step-by-step explanation:
I guessed based on what the side looks like so if it's wrong I"m so so sorry
Fastttt I have 2 more mins please someone answer
Answers
Answer:
Step-by-step explanation:
is 50
360-230=130
180-130=50
The two triangles are similar. Find the values of the unknown variables.
63
B
59°
Р
R
30
C
A
40
Answers
Answer:
x=84
y=59
Step-by-step explanation:
if the triangles are similar than the angles don't change so and y is equal to 59.
to find the missing side we do 40/30 x 63 and we get that x=84
"Matlab
The gradient method was used to find the minimum value of the
function north
f(x,y)=(x^2+y^2−12x−10y+71)^2 Iterations start at the point
(x0,y0)=(2,2.6) and λ=0.002 is used. (The number λ"
Answers
1) The first iteration, n, turns out to be (x1, y1) = ( , ).
2) If the second iteration, n, is (x2, y2) = ( , ).
To find the values of (x1, y1) and (x2, y2), we need additional information or the specific steps of the gradient method applied in MATLAB. The gradient method is an optimization algorithm that iteratively updates the variables based on the gradient of the function. Each iteration involves calculating the gradient, multiplying it by the learning rate (λ), and updating the variables by subtracting the result.
3) After s many iterations (and perhaps changing the value of λ to achieve convergence), it is obtained that the minimum is found at the point (xopt, yopt) = ( , ).
To determine the values of (xopt, yopt), the number of iterations (s) and the specific algorithm steps or convergence criteria need to be provided. The gradient method aims to reach the minimum of the function by iteratively updating the variables until convergence is achieved.
4) The value of the minimum, once the convergence is reached, will be determined by evaluating the function at the point (xopt, yopt). The specific value of the minimum is missing and needs to be provided.
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the complete question is:
Matlab The Gradient Method Was Used To Find The Minimum Value Of The Function North F(X,Y)=(X^2+Y^2−12x−10y+71)^2 Iterations Start At The Point (X0,Y0)=(2,2.6) And Λ=0.002 Is Used. (The Number Λ Is Also Known As The Size Or Step Or Learning Rate.) 1)The First Iteration N Turns Out To Be (X1,Y1)=( , ) 2)If The Second Iteration N Is (X2,Y2)=( ,
Matlab
The gradient method was used to find the minimum value of the function north
f(x,y)=(x^2+y^2−12x−10y+71)^2 Iterations start at the point (x0,y0)=(2,2.6) and λ=0.002 is used. (The number λ is also known as the size or step or learning rate.)
1)The first iteration n turns out to be (x1,y1)=( , )
2)If the second iteration n is (x2,y2)=( , )
3)After s of many iterations (and perhaps change the value of λ to achieve convergence), it is obtained that the minimum is found at the point (xopt,yopt)=( , );
4)Being this minimum=
The difference between high and low tides along the Maine coast one week is 19 feet on Monday and x feet on Tuesday. Write an expression to show the average difference between the tides for Monday and Tuesday. (will give brainliest,)
Answers
Answer: 19 + × / 2
Step-by-step explanation:
Look for the key words:
Rise AND fall of the tide = Addition (19+x)
To show the AVERAGE = Division (Divide it by 2 because there is Monday and Tuesday)
The average difference between the tides for Monday and Tuesday Expression is 1/2×(19 + x)
Average difference Expression = 1/2 ×(19 + x)
Where:
19 represent the Difference between high and low tides on Monday
x represent Difference between high and low tides on Tuesday
Inconclusion The Average difference Expression is 1/2 ×(19 + x)
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What is the constant rate of change?
Answers
Answer: The constant rate of change is 8.
Help please. I have most of the problem done.
Answers
If f(x) = 2x + 3, what is f(-2)?
Answers
Answer:
- 1
Step-by-step explanation:
f(x) = 2x + 3
f( - 2) = 2( - 2) + 3 = - 1
Based on the angles, this triangle will be identified as a(n)
triangle.
Answers
your hair is 5/16
inch long. Write this length as a decimal.
Answers
Answer:
0.3125
Step-by-step explanation:
in decimal form. Use our fraction to decimal calculator to convert any fraction to a decimal and to know if it is a terminating or a recurring (repeating) decimal.
Answer:
0.3125
Step-by-step explanation: