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2 years ago

Solve the equation y′ + 3y = t + e^(−2t).

1 answer:

4 0

Hello,

I am going to remember:

y'+3y=0==>y=C*e^(-3t)

y'=C'*e^(-3t)-3C*e^(-3t)

y'+3y=C'*e^(-3t)-3Ce^(-3t)+3C*e^(-3t)=C'*e^(-3t) = t+e^(-2t)
==>C'=(t+e^(-2t))/e^(-3t)=t*e^(3t)+e^t
==>C=e^t+t*e^(3t) /3-e^(3t)/9

==>y= (e^t+t*e^(3t)/3-e^(3t)/9)*e^(-3t)+D
==>y=e^(-2t)+t/3-1/9+D
==>y=e^(-2t)+t/3+k

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A student raises her grade average from a 75 to a 90. What was the percent of increase

son4ous [18]

This can be solved algebraically. Before we start, as context, to find a percentage you multiply by the percentage/100.

$75*x=90 \\ x= \frac{90}{75} \\ \\ \boxed{x=1.2}$

This tells us that 90 is 120% of 75. However, your question wants us to find the <em>increase</em>, therefore we don't need that 100.
Therefore, the increase is 20%.

4 0

2 years ago

Steve likes to entertain friends at parties with "wire tricks." Suppose he takes a piece of wire 60 inches long and cuts it into

Alex_Xolod [135]

Answer:

a) the length of the wire for the circle = $(\frac{60\pi }{\pi+4}) in$

b)the length of the wire for the square = $(\frac{240}{\pi+4}) in$

c) the smallest possible area = 126.02 in² into two decimal places

Step-by-step explanation:

If one piece of wire for the square is y; and another piece of wire for circle is (60-y).

Then; we can say; let the side of the square be b

so 4(b)=y

b= $\frac{y}{4}$

Area of the square which is L² can now be said to be;

$A_S=(\frac{y}{4})^2 = \frac{y^2}{16}$

On the otherhand; let the radius (r) of the circle be;

2πr = 60-y

$r = \frac{60-y}{2\pi }$

Area of the circle which is πr² can now be;

$A_C= \pi (\frac{60-y}{2\pi } )^2$

$=( \frac{60-y}{4\pi } )^2$

Total Area (A);

A = $A_S+A_C$

= $\frac{y^2}{16} +(\frac{60-y}{4\pi } )^2$

For the smallest possible area; $\frac{dA}{dy}=0$

∴ $\frac{2y}{16}+\frac{2(60-y)(-1)}{4\pi}=0$

If we divide through with (2) and each entity move to the opposite side; we have:

$\frac{y}{18}=\frac{(60-y)}{2\pi}$

By cross multiplying; we have:

2πy = 480 - 8y

collect like terms

(2π + 8) y = 480

which can be reduced to (π + 4)y = 240 by dividing through with 2

$y= \frac{240}{\pi+4}$

∴ since $y= \frac{240}{\pi+4}$ , we can determine for the length of the circle ;

60-y can now be;

= $60-\frac{240}{\pi+4}$

= $\frac{(\pi+4)*60-240}{\pi+40}$

= $\frac{60\pi+240-240}{\pi+4}$

= $(\frac{60\pi}{\pi+4})in$

also, the length of wire for the square (y) ; $y= (\frac{240}{\pi+4})in$

The smallest possible area (A) = $\frac{1}{16} (\frac{240}{\pi+4})^2+(\frac{60\pi}{\pi+y})^2(\frac{1}{4\pi})$

= 126.0223095 in²

≅ 126.02 in² ( to two decimal places)

4 0

2 years ago

Thomas wants to invite Madeline to a party. He has an 80% chance of bumping into her at school. Otherwise, he’ll call her on the

Ivenika [448]

Answer:

84%

Step-by-step explanation:

The probability of Thomas bumping into her at school is 80%, so the probability of not bumping into her is 100% - 80% = 20%.

If he doesn't bump into her (20% chance), he will call her, and the probability of asking her in this case is 60%, so the final probability of asking her in this case is:

$P_1 = 20\% * 60\% = 12\%$

If he bumps into her (80% chance), the probability of asking her is 90%, so the final probability of asking her in this case is:

$P_2 = 80\% * 90\% = 72\%$

To find the probability of Thomas inviting Madeline to the party, we just have to sum the probabilities we found above:

$P = P_1 + P_2$

$P = 12\% + 72\% = 84\%$

5 0

2 years ago

Dori and Malory are tracking their steps taken as a health goal. Dori leaves her house at 12:00 p.m. and walks at 50 steps per m

gladu [14]

Let n = minutes since 12:00 pm when Malory catches up to Dori.

Dori travels
(50 steps/min)*(n minutes) = 50n steps

Malory begins walking at 12:20 pm, so she walks for (n - 20) minutes. She travels
(90 steps/min)*(n - 20 min) = 90n - 1800 steps

Equate the steps traveled by Dori and Malory.
90n - 1800 = 50n
40n = 1800
n = 45 min

The time corresponding to n = 45 min is 12:45 pm

Answer: 12:45 pm

7 0

2 years ago

A multiple choice question has 18 possible answers, only one of which is correct. Is it "significant" to answer a question cor

yan [13]

Answer:

It is not ''significant''

Step-by-step explanation:

Let's call the event

A : ''Answer a question correctly if a random guess is made''

Now we calculate the probability for the event A

$P(A)=\frac{cases where A occurs}{total cases}$