Marcus rented a movie for $4 and some video game for $6 each. He paid $22. How many games did he rent?

If y=e5t is a solution to the differential equation

a_sh-v [17]

Answer:

k = 30, $y(t) = C_1e^{5t}+C_2e^{6t}$

Step-by-step explanation:

Since $y=e^{5t}$ is a solution, then it must satisfy the differential equation. So, we calculate the derivatives and replace the value in the equation. We have that

$\frac{d^2y}{dt^2} = 25 e^{5t},\frac{dy}{dt} = 5e^{5t}$

Then, replacing the derivatives in the equation we have:

$25e^{5t}-11(5)e^{5t}+ke^{5t}=0 e^{5t}(25-55+k) =0$

Since $e^{5t}$ is a positive function, we have that

$25-55+k = 0 \rightarrow k = 30$ .

Now, consider a general solution $y(t) = Ae^{rt}, A \in \mathbb{R}$ , then, by calculating the derivatives and replacing them in the equation, we get

$Ae^{rt}(r^2-11r+30)=0$

We already know that r=5 is a solution of the equation, then we can divide the polynomial by the factor (r-5) to the get the other solution. If we do so, we get that (r-6)=0. So the other solution is r=6.

Therefore, the general solution is

$y(t) = C_1e^{5t}+C_2e^{6t}$

8 0

3 years ago

Indicate in standard form the equation of the line passing through the given points G (4,6) H (1,5)

Delicious77 [7]

First, find the slope (m) = $\frac{y2 - y1}{x2 - x1}$ = $\frac{5 - 6}{1 - 4}$ = $\frac{-1}{-3}$ = $\frac{1}{3}$

Now plug in ONE of the points and the slope into the point-slope equation:

y - y₁ = m(x - x₁); where (x₁, y₁) is the chosen point.

y - 5 = $\frac{1}{3}$ (x - 1) (I used (1,5) as the chosen point)

3(y - 5) = x - 1

3y - 15 = x - 1

3y -14 = x

-14 = x - 3y → x - 3y = -14

Answer: x - 3y = -14

4 0

3 years ago

According to the empirical (69-95-99.7) rule, if a random variable z has a standard normal distribution, then approximately 95%

Dmitry_Shevchenko [17]

Using z-scores, it is found that the value of z is z = 1.96.

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Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula, which for a measure X, in a distribution with mean $\mu$ and standard deviation $\sigma$ , is given by:

$Z = \frac{X - \mu}{\sigma}$

It measures how many standard deviations the measure is from the mean.
Each z-score has an associated p-value, which is the percentile.

The normal distribution is symmetric, which means that the middle 95% is between the <u>2.5th percentile and the 97.5th percentile</u>.
The 2.5th percentile is Z with a p-value of 0.025, thus Z = -1.96.
The 97.5th percentile is Z with a p-value of 0.975, thus Z = 1.96.
Thus, the value of Z is 1.96.

A similar problem is given at brainly.com/question/16965597

7 0

2 years ago

Graph the first six terms of a sequence where a1 = −10 and d = 3.

Neko [114]

notice the first term is -10, and the common difference is 3, so we add 3 to get the next term.

$\bf \begin{array}{cccll}\stackrel{x}{n}&term&\stackrel{y}{value}\\\text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\1&a_1&-10\\\\2&a_2&\stackrel{-10+3}{-7}\\\\3&a_3&\stackrel{-7+3}{-4}\\\\4&a_4&\stackrel{-4+3}{-1}\\\\5&a_5&\stackrel{-1+3}{2}\\\\6&a_6&\stackrel{2+3}{5}\end{array}$