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1 year ago

What is the solution(s) to the system of equations y = x² +4 and y = 2x + 4

Mathematics

1 answer:

bija089 [108]1 year ago

3 0

Answer:

$(x,y) = (0,4)~ \text{and}~ (x,y) = (2,8)$

Step by step explanation:

$y = x^2 +4~~~~~~~~~...(i)\\\\y = 2x +4~~~~~~~~~...(ii)\\\\\text{From (i) and (ii):}\\\\~~~~~~x^2 +4 = 2x+4\\\\\implies x^2 = 2x\\\\\implies x^2 -2x = 0\\\\\implies x(x-2) = 0\\\\\implies x =0,~ x = 2$

$\text{Substitute x = 0 in eq (i):}\\\\y = 0^2 +4 = 4\\\\\text{Substitute x = 2 in eq (i):}\\\\y=2^2+4 = 4+4 = 8\\\\\text{Hence,}~ (x,y)= \{(2,8), (0,4)\}$

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The answer is 8069 insects

P(t) = A * e^(r*t)
P(t) - <span>the insect population at the end of t weeks.
A - the initial number of insects
r - the growth rate
t - number of weeks

P(t) = ?
A = 20
r = 50% = 0.5
t = 12 weeks

P(t) = </span>20 * e^(0.5*12)
P(t) = 20 *
P(t) = 20 * e^(0.5*12)
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3 years ago

Clarissa's sister makes weekly installment payments for a motorized scooter she purchased from a friend. She purchased the motor

Triss [41]

Answer: 18.75x + 150 = 600; Clarissa’s sister will have $150 left to pay after 24 weeks.

Step-by-step explanation: The initial cost is 600 dollars and she has a payment plan of 18.75 dollars per week. This means she has 18.75 times x number of weeks to pay off the entire 600 dollars. Her payment plan can be expressed as follows;

18.75x = 600

However, she wants to make her weekly payments such that at a point she would be left with an outstanding payment of 150 dollars. In other words she wants to know when her weekly payment plus 150 would all be a total of 600. That is 18.75x plus 150 would be equal to 600. Our new equation based on this information is as follows;

18.75x + 150 = 600

18.75x = 600 - 150

18.75x = 450

Divide both sides of the equation by 18.75

x = 24

Hence after paying 18.75 dollars for 24 weeks (450 dollars) she would still have 150 dollars left unpaid.

Therefore the correct equation is;

18.75x + 150 = 600

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3 years ago

-2 = -5/6 (2) +b solve for b

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Answer:

-2 = -5/6 (2) +b

Step-by-step explanation:

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3 years ago

A mass weighing 16 pounds stretches a spring (8/3) feet. The mass is initially released from rest from a point 2 feet below the

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Answer with Step-by-step explanation:

Let a mass weighing 16 pounds stretches a spring $\frac{8}{3}$ feet.

Mass= $m=\frac{W}{g}$

Mass= $m=\frac{16}{32}$

$g=32 ft/s^2$

Mass,m= $\frac{1}{2}$ Slug

By hook's law

$w=kx$

$16=\frac{8}{3} k$

$k=\frac{16\times 3}{8}=6 lb/ft$

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A damping force is numerically equal to 1/2 the instantaneous velocity

$\beta=\frac{1}{2}$

Equation of motion :

$m\frac{d^2x}{dt^2}=-kx-\beta \frac{dx}{dt}+f(t)$

Using this equation

$\frac{1}{2}\frac{d^2x}{dt^2}=-6x-\frac{1}{2}\frac{dx}{dt}+10cos(3t)$

$\frac{1}{2}\frac{d^2x}{dt^2}+\frac{1}{2}\frac{dx}{dt}+6x=10cos(3t)$

$\frac{d^2x}{dt^2}+\frac{dx}{dt}+12x=20cos(3t)$

Auxillary equation

$m^2+m+12=0$

$m=\frac{-1\pm\sqrt{1-4(1)(12)}}{2}$

$m=\frac{-1\pmi\sqrt{47}}{2}$

$m_1=\frac{-1+i\sqrt{47}}{2}$

$m_2=\frac{-1-i\sqrt{47}}{2}$

Complementary function

$e^{\frac{-t}{2}}(c_1cos\frac{\sqrt{47}}{2}+c_2sin\frac{\sqrt{47}}{2})$

To find the particular solution using undetermined coefficient method

$x_p(t)=Acos(3t)+Bsin(3t)$

$x'_p(t)=-3Asin(3t)+3Bcos(3t)$

$x''_p(t)=-9Acos(3t)-9sin(3t)$

This solution satisfied the equation therefore, substitute the values in the differential equation

$-9Acos(3t)-9Bsin(3t)-3Asin(3t)+3Bcos(3t)+12(Acos(3t)+Bsin(3t))=20cos(3t)$

$(3B+3A)cos(3t)+(3B-3A)sin(3t)=20cso(3t)$

Comparing on both sides

$3B+3A=20$

$3B-3A=0$

Adding both equation then, we get

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Now, the general solution

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From initial condition

x(0)=2 ft

x'(0)=0

Substitute the values t=0 and x(0)=2

$2=c_1+\frac{10}{3}$

$2-\frac{10}{3}=c_1$

$c_1=\frac{-4}{3}$

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