The Fourier transform of function [tex]\(f(t)\)[/tex] is given by: [tex]\[F(\omega) = \frac{-(1/4 + j\omega)}{1/16 + \omega^2}\][/tex]
To find the Fourier transform of the function [tex]\(f(t)\)[/tex] given by:
[tex]\[f(t) = \begin{cases} e^{-t/4} & \text{for } t > 0 \\0 & \text{otherwise} \\\end{cases}\][/tex]
The Fourier transform of [tex]\(f(t)\)[/tex] is defined as:
[tex]\[F(\omega) = \int_{-\infty}^{\infty} f(t) \cdot e^{-j\omega t} dt\][/tex]
Let's calculate the Fourier transform using this definition:
For
[tex]\(t > 0\), \(f(t) = e^{-t/4}\)\[F(\omega) = \int_{0}^{\infty} e^{-t/4} \cdot e^{-j\omega t} dt\][/tex]
We can simplify this integral by factoring out the common exponential term:
[tex]\[F(\omega) = \int_{0}^{\infty} e^{-(1/4 + j\omega) t} dt\][/tex]
Now, we can evaluate this integral:
[tex]\[F(\omega) = \left[ \frac{e^{-(1/4 + j\omega) t}}{-(1/4 + j\omega)} \right]_{0}^{\infty}\][/tex]
To evaluate the integral, we consider the limit as t approaches infinity:
[tex]\[F(\omega) = \lim_{t\to\infty} \frac{e^{-(1/4 + j\omega) t}}{-(1/4 + j\omega)} - \frac{e^{-(1/4 + j\omega) \cdot 0}}{-(1/4 + j\omega)}\][/tex]
Since [tex]\(e^{-(1/4 + j\omega) \cdot 0} = e^0 = 1\)[/tex], the second term simplifies to:
[tex]\[F(\omega) = \lim_{t\to\infty} \frac{e^{-(1/4 + j\omega) t} - 1}{-(1/4 + j\omega)}\][/tex]
Now, let's simplify the expression further. We can multiply the numerator and denominator by the complex conjugate of the denominator to rationalize it:
[tex]\[F(\omega) = \lim_{t\to\infty} \frac{e^{-(1/4 + j\omega) t} - 1}{-(1/4 + j\omega)} \cdot \frac{(-1/4 + j\omega)}{(-1/4 + j\omega)}\]\[F(\omega) = \lim_{t\to\infty} \frac{(e^{-(1/4 + j\omega) t} - 1)(-1/4 + j\omega)}{1/16 + \omega^2}\][/tex]
Now, we can evaluate the limit. As t approaches infinity, the exponential term [tex]\(e^{-(1/4 + j\omega) t}\)[/tex] will tend to zero if the real part of the exponent [tex]\(-1/4\)[/tex] is negative. Therefore, we need [tex]\(-1/4 < 0\)[/tex] which is true.
Using L'Hopital's rule, we can differentiate the numerator and denominator with respect to t:
[tex]\[F(\omega) = \frac{\lim_{t\to\infty} \left[ \frac{d}{dt} (e^{-(1/4 + j\omega) t} - 1)(-1/4 + j\omega) \right]}{1/16 + \omega^2}\]\[F(\omega) = \frac{\lim_{t\to\infty} \left[ -(1/4 + j\omega) e^{-(1/4 + j\omega) t} \right]}{1/16 + \omega^2}\][/tex]
Finally, simplifying the expression further, we get:
[tex]\[F(\omega) = \frac{-(1/4 + j\omega)}{1/16 + \omega^2}\][/tex]
Therefore, the Fourier transform of [tex]\(f(t)\)[/tex] is given by:
[tex]\[F(\omega) = \frac{-(1/4 + j\omega)}{1/16 + \omega^2}\][/tex]
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the perimeter of the triangle is ___ units.
Answer:
29 units
Step-by-step explanation:
12 + 12 + 5
29
HELP ASAP PLZ!!! Question in picture!
Answer:YOUR ANSWER IS C
Step-by-step explanation:
Answer:
y=11/2x-10
20 character minimum
question is on the picture
Answer:
12/5
Step-by-step explanation:
tan A = opp/adj
For angle theta, 12 is the opposite leg.
13 is the hypotenuse.
We were not given the length of the adjacent leg, but we can use the Pythagorean theorem to find it.
a^2 + b^2 = c^2
a^2 + 12^2 = 13^2
a^2 + 144 = 169
a^2 = 25
a = 25
The adjacent leg has a length of 5.
tan theta = opp/adj
tan theta = 12/5
Answer: 12/5
Can someone help me out with this please. Do not give me a link for it just answer it please. I will give 20 points.
Answer:
3x+4y=3
Step-by-step explanation:
Plug this in desmos graphing calculator.
PLEASE ANSWER QUICKLY
Answer:
x = 90 and 20 x 4 = 6x9 = k2 over -2
Two parallel lines, I and m, are cut by a transversal, t, as shown below.
1
(17x + 14)
((48-2°
m
What is the value of z?
Answer:
x = 8
Step-by-step explanation:
alternate-interior angles equal 180°
17x + 14 + 4x - 2 = 180
21x + 12 = 180
21x = 168
x = 8
I need help with math.
Step-by-step explanation:
Circumference of a circle= 2πr
where r=6
and π=22/7
C=2πr
=2×22/7×6
=44/7×6
=264/7
=37.71cm
Points that two lines pass through are given in the table. Match each point of intersection to the correct pair of lines.
(2, 0)
(-1, 3)
(1, 0)
(-1, -3)
(0, 2)
(-3, -1)
Answer:
(2,0)(0,2
(-1,-3)(-3,-1)
(-1,3) (1,0)
Step-by-step explanation:
The top of a pie container is a circle. The radius of the pie is 8 inches and the radius of the container is 9 inches. What is the difference between the circumference of the container and the circumference of the pie in inches to the nearest tenth?
A 3.1 inches
B 6.3 inches
C 12.6 inches
D 25.1 inches
E 50.2 inches
Answer:
B
Step-by-step explanation:
2 pi r subtracted by 2 pi r, you get answer
2 x 3.14 ~ 6.28, x 9 ~ 56.54.
2 x 3.14 ~ 6.28, x 8 ~ 50.24
56.54 - 50.24 ~ 6.3
Find the missing side lengths. Leave your answers as radicals in simplest form
Answer:
i think the answer is c but im not sure sorry if im wrong
Step-by-step explanation:
plzz hurryyyy its for my mathhhh classss
Answer:
A
Step-by-step explanation:
4x+2 = 10
=> 2x+2 = 10-2x
please help easy math!! 8th grade math Storypromblem answer number 9
Answer: Chris's Gym pays $2.75
Step-by-step explanation:
So i was glancing at this question wondering how I was gonna figure it out then flipped it upside down in my head and figred out what I was missing
So basically we know that Tyrell pays $2.00 for court reservations
Chris on the other hand according to the table pays $52.75 for court reservation in addition to his monthly fee. Now we have to find out how much is just the court fee. So if we look at the next box I realized that a second court recomendation costs $55.5 and I realized that to find the court reservation costs I needed to subtract so 55.5-52.755= 2.75 which means Chis has a monthly costs of 50 and pays $2.75 for a court holding
If I have 2(2.5)³, is it 2.5 to the power of 3? Or is it 2 to the power of 3?
PLEASE HELP ME OUT AND MAKE SURE TO SHOW YOUR WORK!!!! PLEASEEEEE
Answer:
1. 15
2. 16
3. 46
4. 59
Step-by-step explanation:
1. 44+30=74, 89-74=15
2. 90-74=16
3. 180-134=46
4. angles like that are the same
Answer:
1. x=5
2. x=16
3.x=46
4.x=59
Step-by-step explanation:
1. It says that the angle in total is 89 degrees (faded). then it shows the degrees 44 and 30. add them together to get 74 degrees. next subtract 89 by 74 to get the answer 5.
2. it shows a box at the corner of the angle (box means its a perfect 90 degree angle) so you need to subtract 90 by 74 to get the answer 16.
3. the whole entire angle is 180. so you need to subtract 134 from 180 to get the answer 46.
4. look focus on the one line that shows 59 degrees and right above it there is a blank. that arrow line represents 180 degrees. subtract 180 by 59 to get 121. next focus on the second line (the one that faces downward to the right) you then subtract 180 by 121 to get 59.
1. 44+30=74 89-74=5 x=5
2. box=90 90-74=16 x=16
3. 180-134=46 x=46
4. 180-59=121 180-121=59 x=59
Which of the following equations has a graph that passes through (3, 3) and has a slope of -2?
A. x + 2y = 9
o B. X- 2y = -3
O
C. 2x + y = 9
o
D. -2x + y = -3
Answer:
C
Step-by-step explanation:
y=mx+b
3 = -2(3) + b
3 = -6 + b
9 = b
y = -2x + 9
2x + y = 9
Sam's fish tank has a base area of 187in.? and a height of 16 inches. His fish tank is shown.
16 in.
A = 187 in 2
The tank is currently full of water. How many cubic inches of water should be added to completely fill the tank?
3
® 997 in.
2
B 1,994-
4,488in.3
D 2,992in.
Answer:
D. 2992 cubic inches of water
Step-by-step explanation:
Given data
Base Area= 187in^2
Height of 16 inches
The expression for the volume of a rectangular shaped tank is
V= Area*Height
substiute
V= 187*16
V=2992 in^3
Hence the volume is 2992 cubic inches of water
For the function f(x) = 2.02 – 42 +2 (a) Find the equation for its tangent at r = 0. (b) Find the equation for its tangent at I = 1. (c) Find the equation for its tangent at x = 2. (d) What is special about f when x = 1?
a) The equation for the tangent at x = 0 is y - 2 = -42x.
(b) The equation for the tangent at x = 1 is y + 38 = -38x + 38.
(c) The equation for the tangent at x = 2 is y + 62 = -34x + 68.
(d) When x = 1,the value of f(x) is -38. This is the y- coordinate of the point on the tangent line at x = 1.
What is the explanation for the above ?To find the equations for the tangents of the function f (x) = 2x² – 42x + 2 at specific points,we need to calculate the derivative of the function first. Let's differentiate f(x) with respect to x .
f(x ) = 2x² – 42x+ 2
f'(x) = d/ dx (2x²) – d/dx (42x)+ d/dx (2)
f'(x) = 4x – 42
So
(a) Tangent at x = 0 -
To find the equation for the tangent at x = 0 we need to evaluate f'(x) at x = 0.
f'(0) = 4(0) – 42
f'(0) = -42
Since the slope of the tangent is -42 at x = 0, the equation of the tangent can be written in point-slope form using the point (0, f(0)).
Using f(0) = 2(0)² – 42(0) + 2 = 2, the equation for the tangent at x = 0 is -
y - 2 = -42(x - 0)
y - 2 = -42x
(b) Tangent at x = 1:
To find the equation for the tangent at x = 1,we need to evaluate f'(x) at x = 1.
f'(1 ) =4(1) – 42
f'(1) = -38
Using f(1) = 2(1)² – 42(1) + 2 = -38, the equation for the tangent at x = 1 is -
y + 38 = -38(x - 1)
y + 38 = -38x + 38
(c) Tangent at x = 2
To find the equation for the tangent at x = 2,we need to evaluate f'(x) at x = 2.
f'(2) = 4(2) – 42
f'(2) = -34
Using f(2) = 2(2)^2 – 42(2) + 2 = -62, the equation for the tangent at x = 2 is:
y + 62 = -34(x - 2)
y + 62 = -34x + 68
(d) To determine what is special about f when x = 1, we can evaluate the function at x = 1
f(1) = 2(1)² – 42(1) + 2
f(1) = -38
When x = 1,the value of f(x) is -38. This is the y -coordinate of the point on the tangent line at x = 1. Therefore, the special property of f when x = 1 is that the function value is -38 at that point.
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The total number of passengers riding a certain city bus during the morning shift is 1,000. If the child's fare is $0.50, the adult fare is $1.75, and the total revenue from the fares in the morning shift is $1,500, how many children and how many adults rode the bus during the morning shift?
Let the number of children who rode the bus be C and the number of adults who rode the bus be A. Here's how to solve this problem:
Step-by-step explanation: We are given the following information:
Total number of passengers = 1000 Children's fare = $0.50. Adult fare = $1.75. Total revenue from fares = $1500. Let C be the number of children who rode the bus during the morning shift. Therefore, the number of adults who rode the bus will be (1000 - C). The total revenue from fares is given as:$0.50 × C + $1.75 × (1000 - C) = $1500. Simplifying this equation gives us: 0.5C + 1750 - 1.75C = 1500-1.25C = -250C = 200. Therefore, the number of children who rode the bus is C = 200. Therefore, the number of adults who rode the bus is:1000 - C = 1000 - 200 = 800.
Therefore, the number of adults who rode the bus during the morning shift is 800.
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Adults in various groups were they voted in recent section. The wind be. ARE Yotel der 30 30 50 Over 30 Yes 16 77 27 Nie 24 23 11 Alest is conduct the 0.090 kcalve to whether and voting independent Finthectable for this independence et Ae 10 der Over the value of the forthco AX Which the cont concerned OM-0.050 The water The wash 0.006 the A random sample of 950 Democrats included 817 that consider protecting the environment to be a top priority. A random sample of 800 Republicans included 384 that consider protecting the environment to be a top priority. Construct a 90% confidence interval estimate of the overall difference in the percentages of Democrats and Republicans that prioritize protecting the environment. (Give your answers as percentages, rounded to the nearest tenth of a percent.) Answers The margin of error is 3. We are 90% confident that the difference between the percentage of Democrats and Republicans who prioritize protecting the environment lies between % and 9
The 90% confidence interval estimate for the overall difference in the percentages of Democrats and Republicans who prioritize protecting the environment is between X% and Y%.
To estimate the overall difference in the percentages of Democrats and Republicans who consider protecting the environment a top priority, we can use the concept of confidence intervals.
Confidence intervals provide a range of values within which we can be reasonably confident that the true population parameter lies. In this case, we want to estimate the difference in the proportions of Democrats and Republicans who prioritize protecting the environment.
The provided information states that in a random sample of 950 Democrats, 817 consider protecting the environment a top priority, while in a random sample of 800 Republicans, 384 share the same view. To construct a 90% confidence interval, we need to calculate the margin of error, which is determined by the sample sizes and proportions.
Using the formulas for the margin of error, we find that the margin of error is 3. With this information, we can state that we are 90% confident that the difference between the percentage of Democrats and Republicans who prioritize protecting the environment lies between X% and Y%.
The confidence interval estimate gives us a range within which we can reasonably expect the true difference to fall. However, it does not provide an exact value for the difference. The range allows for some uncertainty due to the sampling variability inherent in survey data.
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URGENT , PLEASE HELP DONT LINK FILES!!!
Find the volume of right rectangular prism made of one hundred 1/2-inch cubes
Answer:
50
Step-by-step explanation:
100÷1/2=50 so your answer would be 50
2. (a) What are the possible remainders when n² + 16n + 20 is divided by 11? (b) Prove for every n € Z that 121 | n² + 16n +20.
The required answer is -
a. leaves a remainder of n² - 3n or (n - 6)(n + 7) ,the possible remainders are 0, 1, 4, 9, 5, 3
b .every n € Z that 121 | n² + 16n +20.
Explanation:-
(a) Remainder when n² + 16n + 20 is divided by 11:Let us first find out the value of n² + 16n + 20,n² + 16n + 20 = (n + 10)(n + 6) + 4As n and 11 are co-prime, by Fermat's Little Theorem, n¹⁰ ≡ 1(mod 11)So, (n + 10)(n + 6) ≡ (n + 1)(n + 7) (mod 11)Hence, n² + 16n + 20 ≡ (n + 1)(n + 7) + 4 ≡ n² + 8n + 11 ≡ n² - 3n (mod 11).Therefore, when n² + 16n + 20 is divided by 11, it leaves a remainder of n² - 3n or (n - 6)(n + 7).
Thus, the possible remainders are 0, 1, 4, 9, 5, 3
(b) Prove that 121 | n² + 16n + 20 for every n ∈ Z . n ∈ Z, thenn² + 16n + 20 = (n + 8)² - 44(n + 8) + 324 = (n + 8 - 22)(n + 8 + 14) + 324= (n - 14)(n + 22) + 324.We know that 121 = 11² | (n - 14)(n + 22),
Therefore, 11 | (n - 14) or 11 | (n + 22)So, there exist k and l ∈ Z such that n - 14 = 11k or n + 22 = 11lWe have, n + 22 = n - 14 + 36Hence, n - 14 ≡ n + 22 (mod 11)If 11 | (n - 14), then 11 | (n + 22), if 11 | (n + 22), then 11 | (n - 14).Therefore, 121 | n² + 16n + 20. Thus, proved.
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The given quadratic expression is: $n^2 + 16n + 20$.
(a)We have to determine the possible remainders when this expression is divided by 11.
Let's solve for the remainder:$(n^2 + 16n + 20) \div 11$ $= \frac{(n + 8)^2 - 44}{11}$
We can write the given expression as $n^2 + 16n + 20 = (n + 8)^2 - 44$
Therefore,$\begin{aligned} (n^2 + 16n + 20) \div 11 & = \frac{(n + 8)^2 - 44}{11} \\ & = \frac{(n + 8)^2}{11} - 4 \end{aligned}$
For all possible values of $n$,
let's calculate the remainder of $\frac{(n + 8)^2}{11}$.
Let $a$ be the remainder when $n$ is divided by 11; then, $(n + 8) \div 11$ has a quotient of $q$ and a remainder of $r$. Therefore, $n + 8 = 11q + r$ $ \Right arrow n = 11q + r - 8$$\begin{aligned} n^2 & = (11q + r - 8)^2 \\ & = 121q^2 + r^2 + 64 + 22qr - 16q - 16r \end{aligned}$Let's now replace $n^2$
in the given expression:$(n^2 + 16n + 20) \div 11 = \frac{121q^2 + r^2 + 22qr - 16q + 6r + 64}{11}$
We have to find the remainders when $121q^2 + r^2 + 22qr - 16q + 6r + 64$ is divided by 11.
We observe that $121q^2$ is always divisible by 11, so we can ignore it.
We must find the remainders of $r^2 + 22qr + 6r + 64 - 16q$. $\begin{aligned} r^2 + 22qr + 6r + 64 - 16q & = r^2 + 2 \cdot 11qr + 11qr + 6r + 64 - 16q \\ & = (r + 11q)^2 + 6r - 16q + 64 \\ & = (r + 11q)^2 - 11(2q - r - 3) \end{aligned}$
The remainder is $r_0 = 11(2q - r - 3)$.
This remainder can take any value between $-10$ and $10$.
(b)We have to show that $121 | n^2 + 16n + 20$ for all integers $n$.We proved earlier that $n^2 + 16n + 20 = (n + 8)^2 - 44$.
We can restate this as $121 | (n + 8)^2 - 44$, which implies that $(n + 8)^2 - 44 = 121k$, where $k \in \mathbb{Z}$.Now we need to show that $k = \frac{n^2 + 16n + 20 - 121k}{121}$ is an integer.
Let's substitute $(n + 8)^2 - 44 = 121k$ into the given expression:$\begin{aligned} k & = \frac{(n + 8)^2 - 44}{121} \\ & = \frac{n^2 + 16n + 20 - 121k}{121} \\ & = \frac{n^2 + 16n + 20}{121} - k \end{aligned}$Thus, $k = \frac{n^2 + 16n + 20 - 121k}{121}$ is an integer for all $n \in \math bb{Z}$,
implying that $121 | n^2 + 16n + 20$ for all $n \in \math bb{Z}$.
Hence, we have proved that $121 | n^2 + 16n + 20$ for all $n \in \math bb{Z}$ using the method of mathematical induction.
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Find the area of the following
triangle:
5 cm
10 cm
A= [?] cm?
Answer:
25
Step-by-step explanation:
A=1/2×b×h
A=1/2×5cm×10cm
A=25cm
Compare the probability that a student will pass the test in the morning with
the probability that a student will pass the test in the afternoon. Draw a
conclusion based on your results.
A. P(pass morning) = 0.24
P(pass afternoon) = 0.41
Conclusion: A student taking the test in the morning has a greater
chance of passing it than a student taking it in the afternoon
O B. P(pass morning) = 0.24
P(pass | afternoon) = 0.41
Conclusion: A student taking the test in the afternoon has a
greater chance of passing it than a student taking it in the
morning.
O c. P(pass morning) = 0.48
P(pass | afternoon) = 0.82
Conclusion: A student taking the test in the afternoon has a
greater chance of passing it than a student taking it in the
morning.
D. P(pass morning) = 0.48
P(pass afternoon) = 0.82
Conclusion: A student taking the test in the morning has a greater
chance of passing it than a student taking it in the afternoon.
Answer: C
Step-by-step explanation:
just took the quiz
The conclusion based on the statement would be option C. Conclusion: A student taking the test in the afternoon has a greater chance of passing it than a student taking it in the morning.
What is the probability?Probability refers to a possibility that deals with the occurrence of random events.
The probability of all the events occurring need to be 1.
We need to Compare the probability that a student will pass the test in the morning with the probability that a student will pass the test in the afternoon.
The conclusion based on the statement would be;
C. P(pass morning) = 0.48
P(pass | afternoon) = 0.82
Conclusion: A student taking the test in the afternoon has a greater chance of passing it than a student taking it in the morning.
Therefore, the correct option is C.
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If I drive at `70` miles per hour, my journey will take `2.5` hours. How long will my journey take if I drive at `60` miles per hour? Explain your answer
Answer:
It'll take 25 more minutes
Step-by-step explanation:
pls help asap this is too hard helpme
On a standardized exam, the scores are normally distributed with a mean of 120 and
a standard deviation of 10. Find the z-score of a person who scored 115 on the exam
Answer: z= -0.5
Step-by-step explanation: mean m=120 and deviation s=10.
Z = (x-m)/s= (115-120)/10
Answer: z=−1.1
Step-by-step explanation:
z=\frac{x-\mu}{\sigma}
z=
σ
x−μ
z-score formula
z=\frac{104-115}{10}
z=
10
104−115
Plug in values
z=\frac{-11}{10}
z=
10
−11
Subtract
z=-1.1
z=−1.1
Divide
I need help with math!!
*PLEASE NO LINKS*
I will mark you brainiest if you don’t send a link!!
Answer:
#1 - y = 2x
#2 - x is days for sale and y is times viewed
#3 - 80
#4 - 35
Step-by-step explanation:
Identify the figure, name the bases,faces,edges, and vertices
Answer:
It’s a square-based pyramid
base(1) is the square on the bottom- ABCD
Faces(4) are the triangles on the sides eg- AED, DEC, CEB, BCA
There are 8 edges, four being at the bottom around the base and the other four being alone the faces
Finally there are 5 vertices, four being at the corners of the base and the fifth being the tip of the pyramid
Step-by-step explanation:
A discrete random variable X is said to be uniformly distributed over the numbers 1. 2. 3......,n, if
P(X=i) =1/n, i=1,2....,n
Find EX and Var(X).
For a discrete random variable X that is uniformly distributed over the numbers 1, 2, 3, ..., n, the expected value (EX) can be calculated as the average of all possible values of X. The variance (Var(X)) measures the spread or variability of the distribution.
The expected value and variance, we need to use the formulas for the mean and variance of a uniform distribution.
The expected value (EX) for a uniform distribution can be calculated as the average of all possible values of X. In this case, since X is uniformly distributed over the numbers 1, 2, 3, ..., n, the expected value can be found using the formula:
EX = (1 + 2 + 3 + ... + n) / n
Simplifying this expression, we get:
EX = (n + 1) / 2
The variance (Var(X)) of a uniform distribution can be calculated using the formula:
Var(X) = (n^2 - 1) / 12
where n is the total number of possible values.
Therefore, for a discrete random variable X that is uniformly distributed over the numbers 1, 2, 3, ..., n, the expected value is (n + 1) / 2, and the variance is (n^2 - 1) / 12. These formulas allow us to calculate the average and measure the spread of the distribution.
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Tony will spend at most $27 on gifts. So far, he has spent $14. What are the possible additional amounts he will spend?
Use c for the additional amount (in dollars) Tony will spend.
Write your answer as an inequality solved for c.
Answer:
Inequaltiy: 27 ≤ 14+c
Final answer: 13 ≤ c
Step-by-step explanation:
Well tony spent at most 27 dollars.
He spent 14 already and he can‘t get past 27 dollars but he can spend exactly 27 or even less.
So 27 ≤ 14+c