Mathematics, 02.06.2021 03:40 Animallover100

A factory is discharging pollution into a lake at the rate of r(t) tons per year given below, where t is the number of years that the factory has been in operation. Find the total amount of pollution discharged during the first 7 years of operation. (Round your answer to two decimal places.)

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Answer from: Roof55

The total amount of pollution discharged during the first 7 years of operation is 1.955 tons

Step-by-step explanation:

Given

$r(t) = \frac{t}{t^2 + 1}$

Required

The total amount in the first 7 years

This implies that:

$r(t) = \frac{t}{t^2 + 1}; [0,7]$

The total amount is calculated by integrating r(t) i.e.

$v = \int\limits^a_b {r(t)} \, dt$

So:

$v = \int\limits^7_0 {\frac{t}{t^2 + 1}} \, dt$

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

We have:

t^2 + 1

Differentiate

d(t^2 + 1) = 2t

Rewrite as:

2t = d(t^2 + 1)

Solve for t

$t = \frac{1}{2}d(t^2 + 1)$

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

So:

Make t the subject

$v = \int\limits^7_0 {\frac{t}{t^2 + 1}} \, dt$

$v = \int\limits^7_0\frac{1}{2}* {\frac{d(t^2 + 1)}{t^2 + 1}} \, dt$

$v = \frac{1}{2}\int\limits^7_0 {\frac{d(t^2 + 1)}{t^2 + 1}} \, dt$

Integrate

$v = \frac{1}{2}\ln(t^2 +1)|\limits^7_0$

Expand

$v = \frac{1}{2}[\ln(7^2 +1) - \ln(0^2 +1)]$

$v = \frac{1}{2}[\ln(50) - \ln(1)]$

$v = \frac{1}{2}[3.91 - 0]$

$v = \frac{1}{2}[3.91]$

v = 1.955

Answer from: Quest

3201

step-by-step explanation:

Answer from: Quest

a. we have two

lines: y = 4-x and

y = 8-x^-1

given two simultaneous equations that are both to be

true, then the solution is the points where the lines cross. the intersection is

where the two equations are equal. therefore the solution that works for both

equations is when

4-x = 8-x^-1

this is where the two lines will cross and that is the

common point that satisfies both equations.

b. 4-x = 8-x^-1

x 4-x 8-x^-1

-3 7

8.33

-2 6 8.5

-1 5 9

0 4

1 3

2 2

7.5

3 1 7.67

the table shows that none of the x values from -3 to 3 is

the solution because in no case does

4-x = 8-x^-1

to find the solution we need to rearrange the equation to

find for x:

4-x = 8-x^-1

multiply both sides with x:

4x-x^2 = 8x-1

x^2+4x-1=0

x= -4.236, 0.236

therefore there are two points that satisfies the

equation.

find y:

x=-4.236

y = 4-x = 4 – (-4.236)

= 8.236

y = 8-x^-1 = .236)^-1

= 8.236

x=0.236

y = 4-x = 4 – (0.236) = 3.764

y = 8-x^-1 = 8-(0.236)^-1

= 3.764

thus the two lines cross at 2 points:

(-4.236, 8.236) & (0.236, 3.764)

c. to solve

graphically the equation 4-x = 8-x^-1

we would graph both lines: y = 4-x and y

= 8-x^-1

the point on the graph where the lines cross is the

solution to the system of equations.

just graph the points on part b on a cartesian coordinate

system and extend the two lines. the

solution is, as stated, the point where the two lines cross on the graph.

step-by-step explanation:

mark brainliest!

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A factory is discharging pollution into a lake at the rate of r(t) tons per year given below, where...