lilyrockstarmag

12.05.2021 •

Mathematics

The width of a rectangle is 4.5 inches and the diagonal length of the rectangle is 7.5 inches. Which measurement is closest to the length of this rectangle in inches? 3 inches
6 inches
9 inches
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Ответ:

goaliekid2663

25.12.2021

Mathematics

(a) $\frac{dy}{dt}=k\left ( 700000-y(t) \right )$

(b) $y(t)=700000-700000e^{\frac{1}{9}\ln \left ( \frac{1}{2} \right )t}$

Step-by-step explanation:

Given: News spreads through a city of fixed size of 700000 people at a time rate proportional to the number of people who have not heard the news.

To find:

(a) a differential equation

(b) number of people who have heard the news after t days

Solution:

(a)

Total number of people in a city = 700000

As y(t) denotes the number of people who have heard the news t days after it has happened

Number of people who have not heard the news = 700000 - y(t)

So, differential equation is $\frac{dy}{dt}=k\left ( 700000-y(t) \right )$

Here, k is the proportionality constant.

(b)

Integrate both sides of the differential equation.

$\frac{dy}{dt}=k\left ( 700000-y(t) \right )\\\int \frac{dy}{\left ( 700000-y(t) \right )}=\int k\,dt\\ln\left ( 700000-y \right )=kt+C\,\,\left \{ \because \int \frac{dy}{y}=\ln y+C \right \}$

Use y(0)=0

$ln\left ( 700000-y \right )=kt+C\\\ln (700000)=C\\\Rightarrow ln\left ( 700000-y \right )=kt+\ln (700000)$

As a poll showed that 350000 people have heard the news 9 days after a scandal in City Hall was reported, y(9)=350000

$ln\left ( 700000-y \right )=kt+\ln (70000)\\ln\left ( 700000-350000 \right )=kt+\ln (700000)\\\ln (350000)=9k+\ln (700000)\\9k=\ln (350000)-\ln (700000)\\$

$9k=\ln \left ( \frac{350000}{700000} \right )\\9k=\ln \left ( \frac{1}{2} \right )\\k=\frac{1}{9}\ln \left ( \frac{1}{2} \right )\\ln\left ( 700000-y \right )=\frac{1}{9}\ln \left ( \frac{1}{2} \right )t+\ln (700000)\\ln\left ( 700000-y \right )-\ln (700000)=\frac{1}{9}\ln \left ( \frac{1}{2} \right )t\\$

$\ln \left ( \frac{700000-y }{700000} \right )=\frac{1}{9}\ln \left ( \frac{1}{2} \right )t\\ \frac{700000-y }{700000}=e^{\frac{1}{9}\ln \left ( \frac{1}{2} \right )t}\\700000-y=700000e^{\frac{1}{9}\ln \left ( \frac{1}{2} \right )t}\\y=700000-700000e^{\frac{1}{9}\ln \left ( \frac{1}{2} \right )t}\\$

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The width of a rectangle is 4.5 inches and the diagonal length of the rectangle is 7.5 inches. Which measurement is closest to the length of this rectangle in inches? 3 inches 6 inches 9 inches Not Here