Simultaneous equations 5x-4y=19
X+2y=8

Answer:

a) The expression for the number of bacteria is [tex]P(t) = 100\cdot e^{1.131\cdot t}[/tex].

b) There are 2975 bacteria after 3 hours.

c) The rate of growth after 3 hours is about 3365.3 bacteria per hour.

d) A population of 10,000 will be reached after 4.072 hours.

Step-by-step explanation:

a) The population growth of the bacteria culture is described by this ordinary differential equation:

[tex]\frac{dP}{dt} = k\cdot P[/tex] (1)

Where:

[tex]k[/tex] - Rate of proportionality, in [tex]\frac{1}{h}[/tex].

[tex]P[/tex] - Population of the bacteria culture, no unit.

[tex]t[/tex] - Time, in hours.

The solution of this differential equation is:

[tex]P(t) = P_{o}\cdot e^{k\cdot t}[/tex] (2)

Where:

[tex]P_{o}[/tex] - Initial population, no unit.

[tex]P(t)[/tex] - Current population, no unit.

If we know that [tex]P_{o} = 100[/tex], [tex]t = 1\,h[/tex] and [tex]P(t) = 310[/tex], then the rate of proportionality is:

[tex]P(t) = P_{o}\cdot e^{k\cdot t}[/tex]

[tex]\frac{P(t)}{P_{o}} = e^{k\cdot t}[/tex]

[tex]k\cdot t = \ln \frac{P(t)}{P_{o}}[/tex]

[tex]k = \frac{1}{t}\cdot \ln \frac{P(t)}{P_{o}}[/tex]

[tex]k = \frac{1}{1}\cdot \ln \frac{310}{100}[/tex]

[tex]k\approx 1.131\,\frac{1}{h}[/tex]

Hence, the expression for the number of bacteria is [tex]P(t) = 100\cdot e^{1.131\cdot t}[/tex].

b) If we know that [tex]t = 3\,h[/tex], then the number of bacteria is:

[tex]P(t) = 100\cdot e^{1.131\cdot t}[/tex]

[tex]P(3) = 100\cdot e^{1.131\cdot (3)}[/tex]

[tex]P(3) \approx 2975.508[/tex]

There are 2975 bacteria after 3 hours.

c) The rate of growth of the population is represented by (1):

[tex]\frac{dP}{dt} = k\cdot P[/tex]

If we know that [tex]k\approx 1.131\,\frac{1}{h}[/tex] and [tex]P \approx 2975.508[/tex], then the rate of growth after 3 hours:

[tex]\frac{dP}{dt} = \left(1.131\,\frac{1}{h} \right)\cdot (2975.508)[/tex]

[tex]\frac{dP}{dt} = 3365.3\,\frac{1}{h}[/tex]

The rate of growth after 3 hours is about 3365.3 bacteria per hour.

d) If we know that [tex]P(t) = 10000[/tex], then the time associated with the size of the bacteria culture is:

[tex]P(t) = 100\cdot e^{1.131\cdot t}[/tex]

[tex]10000 = 100\cdot e^{1.131\cdot t}[/tex]

[tex]100 = e^{1.131\cdot t}[/tex]

[tex]\ln 100 = 1.131\cdot t[/tex]

[tex]t = \frac{\ln 100}{1.131}[/tex]

[tex]t \approx 4.072\,h[/tex]

A population of 10,000 will be reached after 4.072 hours.

Answer:

r = 3 2/3

s = -0.444333

Step-by-step explanation:

Multiply the top equation by 2

14r - 6s = 52

2r -  6s =   8          Subtract the two equations

12r = 44                 Divide by 12

r = 44/12

r = 3 8/12

r = 3 2/3

2r - 6s = 8

2*(2 2/3) - 6s = 8

2*2.6667 - 6s = 8

5.3334 - 6s = 8              Subtract 5.3334 from both sides.

- 6s = 2 2/3                    Divide by - 6

s = - 0.4443333

Answer:

7 apples into 2 pieces and 4 apples into 4 pieces

Step-by-step explanation:

if you split 7 apples into 2 pieces each than you'l have 14 slices. You need 30 though which means you need 16 more. so you split 4 into 4 pieces. and the number of apples we used is 7 and 4 which make up 11. So this answer works

Answer:

No solution

Step-by-step explanation:

5x+10y=3 equation 1

10x+20y=8 equation 2

-2(5x+10y)=-2(3) multiply equation 1 by -2 to eliminate x

-10x-20y=-6 equation 1 multiplied by -2

10x+20y=8 equation 2

0  +  0  =2. Add above equations

    0      =2  

no solution

Answer:

[tex]\displaystyle f(3x) = \frac{6x + 5}{3x + 4}[/tex]

General Formulas and Concepts:

Algebra I

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle f(x) = \frac{2x + 5}{x + 4}[/tex]

Step 2: Find

Answer:

C.

Step-by-step explanation:

For an angle to be supplementary to another angle, they must be equal to 180. Angles 6, 10, 13, and 9 are all supplementary to angle 16. Although there are more choices in different answers it wouldn't work with question, so C is the right answer.

Angle 16 is supplementary to angle 9 by the Same Side Interior Angles Theorem, which makes it supplementary to angle 10 by the Alternate Exterior Angles Theorem, which is also congruent to angle 13 by the Vertical Angles Theorem, which is also supplementary to angle 6 by the Alternate Exterior Angles Theorem.

Answer:

remember the chain rule:

h(x) = f(g(x))

h'(x) = f'(g(x))*g'(x)

or:

dh/dx = (df/dg)*(dg/dx)

we know that:

z = 4*e^x*ln(y)

where:

y = u*sin(v)

x = ln(u*cos(v))

We want to find:

dz/du

because y and x are functions of u, we can write this as:

dz/du = (dz/dx)*(dx/du) + (dz/dy)*(dy/du)

where:

(dz/dx)  = 4*e^x*ln(y)

(dz/dy) = 4*e^x*(1/y)

(dx/du) = 1/(u*cos(v))*cos(v) = 1/u

(dy/du) = sin(v)

Replacing all of these we get:

dz/du = (4*e^x*ln(y))*( 1/u) + 4*e^x*(1/y)*sin(v)

          = 4*e^x*( ln(y)/u + sin(v)/y)

replacing x and y we get:

dz/du = 4*e^(ln (u cos v))*( ln(u sin v)/u + sin(v)/(u*sin(v))

dz/du = 4*(u*cos(v))*(ln(u*sin(v))/u + 1/u)

Now let's do the same for dz/dv

dz/dv = (dz/dx)*(dx/dv) + (dz/dy)*(dy/dv)

where:

(dz/dx)  = 4*e^x*ln(y)

(dz/dy) = 4*e^x*(1/y)

(dx/dv) = 1/(cos(v))*-sin(v) = -tan(v)

(dy/dv) = u*cos(v)

then:

dz/dv = 4*e^x*[ -ln(y)*tan(v) + u*cos(v)/y]

replacing the values of x and y we get:

dz/dv = 4*e^(ln(u*cos(v)))*[ -ln(u*sin(v))*tan(v) + u*cos(v)/(u*sin(v))]

dz/dv = 4*(u*cos(v))*[ -ln(u*sin(v))*tan(v) + 1/tan(v)]

===========================================================

Explanation:

We have these two functions

which represent the amounts for his friend and William in that order. Strangely your teacher mentions William first, but then swaps the order when listing the exponential function as the first. This might be slightly confusing.

The table of values is shown below. We have t represent the number of years and t starts at 0.5. It increments by 0.1

The f(t) and g(t) columns represent the outputs for those mentioned values of t. For example, if t = 0.5 years (aka 6 months) then f(t) = 12.48 and that indicates his friend has 12,480 dollars in the account.

I've added a fourth column labeled |f - g| which represents the absolute value of the difference of the f and g columns. If f = g, then f-g = 0. The goal is to see if we get 0 in this column or try to get as close as possible. This occurs when we get 0.09 when t = 1.0

So we don't exactly get f(t) and g(t) perfectly equal, but they get very close when t = 1.0

It turns out that the more accurate solution is roughly t = 0.9925 which is close enough. I used a graphing calculator to find this approximate solution.

It takes about a year for the two accounts to have the same approximate amount of money.

Answer:

B

Step-by-step explanation:

Answer:

Step-by-step explanation:

P=R3000.00

T=3 years

SI=R905

SI=P\times R\times T\\R905=R3000\times \frac{r}{100}\times 3SI=P×R×T

R905=R3000×  

100

r

​

×3

R905=\frac{R9000r}{100}R905=  

100

R9000r

​

R905=\frac{R905}{R90}R905=  

R90

R905

​

r=\frac{R905}{R90.}r=  

R90.

R905

​

r=10.06\%r=10.06%

Answer:

1:1/3

Step-by-step explanation:

60:20

6:2

1:1/3

n=1/3

Brainliest please~

The value of n=100/3

As per given the value of 1m 100cm

then the ratio of value be 60/2000 is equal to the 1/(2000/60) 1/(100/3) on compare with 1:n then the Value be

n=100/3

In mathematics, a ratio indicates how often one number contains another. For example, if you have 8 oranges and 6 lemons in a fruit bowl, the ratio of oranges to lemons will be 8: 6 (that is, 8: 6, or 4: 3).

For example, if you have one boy and three girls, you can write the ratio as follows: 1: 3 (every boy has 3 girls) 1/4 is a boy and 3/4 is a girl. 0.25 is a boy (by dividing 1 by 4)

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Without any extra conditions, the answer could be 3, and the simplest polynomial with the given roots would be

(x + 5) (x - (1 + 4i )) (x + 4i )

= x ³ + 4x ² + (11 - 4i ) x + 80 - 2i

If the polynomial is supposed to have only real coefficients, then any complex roots must occur along with their complex conjugates:

(x + 5) (x - (1 + 4i )) (x - (1 - 4i )) (x + 4i ) (x - 4i )

= x ⁵ + 3x ⁴ + 23x ³ + 133x ² + 112x + 1360

and then the degree would be 5.

Answer:

The circumference and diameter of a circle

Step-by-step explanation:

Proportional relationships can be written as [tex]y=kx[/tex], where [tex]k[/tex] is some constant of proportionality. The formula for a circumference of a circle can be written as [tex]C=d\pi[/tex], where [tex]d[/tex] is the diameter of the circle. Therefore, the constant of proportionality is [tex]\pi[/tex] and the circumference and diameter of a circle are in a proportional relationship.

Answer:

The numbers are 65, 67, and 69

Step-by-step explanation:

Hi there!

We need to find 3 consecutive odd integers.

Consecutive numbers are numbers that follow each other (ex. 1, 2, 3, 4)

We're given that 5 times the first number + 4 times the second + 3 times the third = 800

Let's make the first number x

Since the second number is consecutive to the first and odd, it will be x+2 (Why? Well, let's say x is 5. In that case, x+1=6, which is even. However, x+2=7)

Therefore, the third number is x+4 (once again, if x is 5, x+3=8, but x+4=9)

5 times the first number is 5x

4 times the second is 4(x+2)

3 times the third is 3(x+4)

And of course, that equals 800

As an equation, it'll be:

5x+4(x+2)+3(x+4)=800

open the parenthesis

5x+4x+8+3x+12=800

combine like terms

12x+20=800

Subtract 20 from both sides

12x=780

Divide by 12 on both sides

x=65

The first number is x, so the first number is 65

The second number is x+2, or 65+2=67

The third number is x+4, or 65+4=69

Hope this helps!

Answer:

Plastic is 22.5 pounds and aluminum is 32.5 pounds.

Step-by-step explanation:

total junk = 55 pounds

Value of plastic = $ 1.60 per pound

Value of aluminum = $ 1.28 per pound

Total value= $ 77.60

Let the plastic is p and the aluminum is 55 - p.

Total cost

77.60 = 1.6 p + (55 - p) x 1.28

77.60 =  1.6 p + 70.4 - 1.28 p

7.2 = 0.32 p

p = 22.5 pounds

So, plastic is 22.5 pounds and aluminum is 32.5 pounds.

Answer:

Step-by-step explanation:

commission = 0.7% of $12,500

= 0.007×$12,500

= $87.5

Answer:

C

Step-by-step explanation:

p=20+1

Answer:

12.259-12.25 890654321

9514 1404 393

Answer:

  0.5

Step-by-step explanation:

The "enclosed area" can be taken to mean different things. Here, we consider it to mean the finite area bounded between the two curves, regardless of which curve is higher value than the other.

The area is bounded on the interval [0, 2]. On half that interval y1 > y2; on the other half, y2 > y1. This means the integral of the area between the curves will be different for one part of the interval than for the other. (The curves are symmetric about the point (1, 0).)

The area on the interval [0, 1] is given by the integral ...

  [tex]\displaystyle\int_0^1{(y_1-y_2)}\,dx=\int_0^1{((x-1)^3-(x-1))}\,dx\\\\=\int^1_0{(x(x-1)(x -2))}\,dx=\left.(\frac{x^4}{4}-x^3+x^2)\right|^1_0=\boxed{\frac{1}{4}}[/tex]

The area on the interval [1, 2] is the integral of the opposite integrand, but has the same value.

The positive area over the whole interval from 0 to 2 is 1/4+1/4 = 1/2.

If you simply integrate y2-y1 or y1-y2 over that interval, the result is 0.

Answer:

x=70

y=30

z=20

Step-by-step explanation:

x=180-110 (angles on a straight line)

y=180-110-40 (angle sum of triangle)

z= 180-90-70 (angle sum of triangle)

Answer:

x=70°

y=30°

z=20°

Step-by-step explanation:

x=180°-110°(anlges on a straight line)

x=70°

y+110°+40°=180°(sum of angles of triangle)

y+150°=180°

y=180°-150°

y=30°

z+x+90°=180°(sum of angles of triangle)

z+70°+90°=180°

z+160°=180°

z=180°-160°

z=20°

Dada una ecuación de la forma

Tenemos que:

Sabemos que:

f(x)=1+6Sen(2x+π/3)

Y podemos reescribirla como:

f(x)=6Sen(2(x+π/6))+1

Siendo:

La imagen de una función trigonométrica de esta forma es:

y ∈ [-A+D,A+D]

y ∈ [-6+1, 6+1]

y ∈ [-5,7]

La gráfica se adjunta.

Answer:

[tex]g(x) = \frac{-2}{x-1}+5\\\\[/tex]

The graph is shown below.

=========================================================

Explanation:

Notice that if we multiplied the denominator (x-1) by 5, then we get 5(x-1) = 5x-5.

This is close to 5x-7, except we're off by 2 units.

In other words,

5x-7 = (5x-5)-2

since -7 = -5-2

Based on that, we can then say,

[tex]g(x) = \frac{5x-7}{x-1}\\\\g(x) = \frac{5x-5-2}{x-1}\\\\g(x) = \frac{(5x-5)-2}{x-1}\\\\g(x) = \frac{5(x-1)-2}{x-1}\\\\g(x) = \frac{5(x-1)}{x-1}+\frac{-2}{x-1}\\\\g(x) = 5+\frac{-2}{x-1}\\\\g(x) = \frac{-2}{x-1}+5[/tex]

This answer can be reached through alternative methods of polynomial long division or synthetic division (two related yet slightly different methods).

-------------------------

Compare the equation [tex]g(x) = \frac{-2}{x-1}+5\\\\[/tex] to the form [tex]g(x) = \frac{a}{x-h}+k\\\\[/tex]

We can see that

The vertical asymptote is x = 1, which is directly from the h = 1 value. If we tried plugging x = 1 into g(x), then we'll get a division by zero error. So this is why the vertical asymptote is located here.

The horizontal asymptote is y = 5, which is directly tied to the k = 5 value. As x gets infinitely large, then y = g(x) slowly approaches y = 5. We never actually arrive to this exact y value. Try plugging in g(x) = 5 and solving for x. You'll find that no solution for x exists.

The point (h,k) is the intersection of the horizontal and vertical asymptote. It's effectively the "center" of the hyperbola, so to speak.

The graph is shown below. Some points of interest on the hyperbola are

Another thing to notice is that this function is always increasing. This means as we move from left to right, the function curve goes uphill.

Answer:

mark me as brinalist if answers are correct

Answer:

D is the answer

Step-by-step explanation:

plug the -2's in line 1 & 2 then 4 in 2 and 3

the 1&2 , and the 2 and 3 numbers have to match

Using the conditions for continuity, we find that the function D.) is continuous.

A function f(x) is said to be continuous at a point x = a, in its domain if the following three conditions are satisfied:

Since for -4 <= x < -2, -2 <= x < 4 and 4 <= x <= 8, the function f(x) is defined by straight lines , the function will be continuous for all x ≠ -2 and 4. Now for x = -2, 4, let us check all the three conditions:

A.

f(-2) = 0.5 * (-2)² = 2

left hand limit = -2 + 6 = 4

right hand limit = 0.5 * (-2)² = 2

Since, left hand limit is not equal to right hand limit, the function is not continuous at x = -2. No need to check further.

B.

f(-2) = 0.5 * (-2)² = 2

left hand limit = -2 -2 = -4

right hand limit = 0.5 * (-2)² = 2

Since, left hand limit is not equal to right hand limit, the function is not continuous at x = -2. No need to check further.

C.

f(-2) = 0.5 * (-2)² = 2

left hand limit = -2 + 4 = 2

right hand limit = 0.5 * (-2)² = 2

Since, left hand limit is not equal to right hand limit is equal to f(-2), the function is continuous at x = -2.

f(4) = 25 - 3*4 = 13

left hand limit = 0.5 * (4)² = 8

right hand limit = 25 - 3*4 = 13

Since, left hand limit is not equal to right hand limit, the function is not continuous at x = 4.

D.

f(-2) = 0.5 * (-2)² = 2

left hand limit = -2 + 4 = 2

right hand limit = 0.5 * (-2)² = 2

Since, left hand limit is not equal to right hand limit is equal to f(-2), the function is continuous at x = -2.

f(4) = 20 - 3*4 = 8

left hand limit = 0.5 * (4)² = 8

right hand limit = 20 - 3*4 = 8

Since, left hand limit is not equal to right hand limit is equal to f(-2), the function is continuous at x = 4.

Thus, the function is continuous.

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Answer:

Infinitely many solutions.

They must satisfy [tex]y = \frac{1}{5}(x - 5)[/tex]

Step-by-step explanation:

Given

[tex]x - 5y = 5[/tex]

[tex]-x + 5y = -5[/tex]

Required

The best description

Add both equations

[tex]x - x - 5y + 5y = 5 - 5[/tex]

[tex]0+0 =0[/tex]

[tex]0 = 0[/tex] ---- this means that the system has infinitely many solutions.

Make y the subject in: [tex]-x + 5y = -5[/tex]

Add x to both sides

[tex]5y = x - 5[/tex]

Divide through by 5

[tex]y = \frac{1}{5}(x - 5)[/tex]

Hence, they must satisfy: [tex]y = \frac{1}{5}(x - 5)[/tex]

Answer/Step-by-step explanation:

✔️-72 ÷ (-2)

The division of two negative numbers will give us a positive number. i.e. - ÷ - = +

Therefore:

-72 ÷ (-2) = 36

✔️-72 ÷ 2

The division of a negative number and a positive number will give us a negative number. i.e. - ÷ + = -

Therefore:

-72 ÷ 2 = -36

✔️72 ÷ (-2)

The division of a positive number and a negative number will give us a negative number. i.e. + ÷ - = -

Therefore:

72 ÷ (-2) = -36

Answer:

i put in 3 to make 23436 because 36 is divisible by 6

Answer:

Jose has 5.2 meters of fabric left.

Step-by-step explanation:

5.6 - 0.4 = 5.2

9514 1404 393

Answer:

  A.  7.2

Step-by-step explanation:

In this geometry, all of the right triangles are similar. This means corresponding sides have the same ratio.

  short side/hypotenuse = x/12 = 12/20

Multiplying by 12 gives ...

  x = 12(12/20) = 144/20

  x = 7.2