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3 years ago

Polygon ABCDE rotates 45° clockwise about point F to produce polygon FGHIJ as shown. What is mBCD?

Mathematics

1 answer:

goldfiish [28.3K]3 years ago

5 0

Polygon FGHIJ is the result of the rotation of ABCDE.

A = F ⇒ 117°
B = G ⇒ 90°
C = H ⇒ 127°
D = I ⇒ 90°
E = J ⇒ 117°

The measure of angle BCD remains the same because the rotation involves the whole polygon. Thus, it is still 127°

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Use the laplace transform to solve the given initial-value problem. y' 5y = e4t, y(0) = 2

Basile [38]

The Laplace transform of the given initial-value problem

$y' 5y = e^{4t}, y(0) = 2$ is mathematically given as

$y(t)=\frac{1}{9} e^{4 t}+\frac{17}{9} e^{-5 t}$

<h3>What is the Laplace transform of the given initial-value problem? y' 5y = e4t, y(0) = 2?</h3>

Generally, the equation for the problem is mathematically given as

$&\text { Sol:- } \quad y^{\prime}+s y=e^{4 t}, y(0)=2 \\\\&\text { Taking Laplace transform of (1) } \\\\&\quad L\left[y^{\prime}+5 y\right]=\left[\left[e^{4 t}\right]\right. \\\\&\Rightarrow \quad L\left[y^{\prime}\right]+5 L[y]=\frac{1}{s-4} \\\\&\Rightarrow \quad s y(s)-y(0)+5 y(s)=\frac{1}{s-4} \\\\&\Rightarrow \quad(s+5) y(s)=\frac{1}{s-4}+2 \\\\&\Rightarrow \quad y(s)=\frac{1}{s+5}\left[\frac{1}{s-4}+2\right]=\frac{2 s-7}{(s+5)(s-4)}\end{aligned}$

$\begin{aligned}&\text { Let } \frac{2 s-7}{(s+5)(s-4)}=\frac{a_{0}}{s-4}+\frac{a_{1}}{s+5} \\&\Rightarrow 2 s-7=a_{0}(s+s)+a_{1}(s-4)\end{aligned}$

$put $s=-s \Rightarrow a_{1}=\frac{17}{9}$$

$\begin{aligned}\text { put } s &=4 \Rightarrow a_{0}=\frac{1}{9} \\\Rightarrow \quad y(s) &=\frac{1}{9(s-4)}+\frac{17}{9(s+s)}\end{aligned}$

In conclusion, Taking inverse Laplace tranoform

$L^{-1}[y(s)]=\frac{1}{9} L^{-1}\left[\frac{1}{s-4}\right]+\frac{17}{9} L^{-1}\left[\frac{1}{s+5}\right]$ \\\\$

$y(t)=\frac{1}{9} e^{4 t}+\frac{17}{9} e^{-5 t}$

Read more about Laplace tranoform

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6 0

1 year ago

Tickets for a school football game cost 1.00 if purchased before the day of the game. They cost 1.50. They cost 1.50 each if bou

cluponka [151]

Answer:

Let x = number of gate tickets, ($1.50)

Since there were 600 tickets sold, let (600-x) = the $1 tickets

1.50 tickets + $1 tickets = $700

1.5x + 1(600 - x) = 700

1.5x - x = 700 - 600

.5x = 100

x = 100/.5

x = 200 ea 1.50 tickets sold at the gate

;

Check: there were 600 - 200 = 400 ea $1 tickets sold

1.50(200) + 1(400) = $700

6 0

3 years ago

The population of bacteria in a certain culture can be modeled by the function P(t)=10000(1.03)^t. Is the population of bacteria

ch4aika [34]

Answer:

The population of bacteria is increasing.

Step-by-step explanation:

The function modelling the population of bacteria in the given culture is an exponential function; We have a base 1.03 and an exponent t. An exponential function is said to be increasing if the base is strictly greater than 1, this implies that the population of the bacteria is increasing as t increases from 0 to infinity.