Someone verify these answers.
Thank you!
1. A biologist finds the average production in which cancerous cells are produced in rats is given by p(t)= (t^2-2t+1)/t where t is measured in hours and p(t) is measured in 1000 cells per hour. Estimate the rate at which the average production is changing at 4 hours.
It's an instantaneous rate of change
so 4 hours
I plug in the 3.99 and 4.01 in the equation (t^2-2t+1)/t
which equals to 0.937 x 1000=937?
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2. State the points where f(x) = x^2 - 8 and the reciprocal of f(x) intersect.
Answer: They intersect at (√9, 1), (-√9, 1) and (-√7, -1), (√7, -1)
I agree with the answers to both of your questions
The only objection would be ..
for (√9,1) and (-√9,1) you should have had
(3,1) and (-3,1)
1. To estimate the rate at which the average production is changing at 4 hours, you can use the concept of the instantaneous rate of change.
First, plug in the values 3.99 and 4.01 into the given equation: p(t) = (t^2 - 2t + 1)/t
So, at t = 3.99 hours: p(3.99) = (3.99^2 - 2(3.99) + 1)/3.99 = 0.937
Therefore, at t = 3.99 hours, the average production is approximately 0.937 multiplied by 1000 cells per hour, which equals 937 cells per hour.
Similarly, at t = 4.01 hours: p(4.01) = (4.01^2 - 2(4.01) + 1)/4.01 = 0.937
So, at t = 4.01 hours, the average production is also approximately 0.937 multiplied by 1000 cells per hour, which equals 937 cells per hour.
Therefore, you can conclude that the estimated rate at which the average production changes at 4 hours is 937 cells per hour.
2. To find the points of intersection between the function f(x) = x^2 - 8 and its reciprocal, you need to set these two equations equal to each other and solve for x.
So, set f(x) = x^2 - 8 equal to 1/f(x).
This means: x^2 - 8 = 1/(x^2 - 8)
To simplify, you can multiply both sides of the equation by (x^2 - 8), which gives you:
(x^2 - 8)(x^2 - 8) = 1
Expanding the left side of the equation:
x^4 - 16x^2 + 64 = 1
Rearranging the equation:
x^4 - 16x^2 + 63 = 0
This is a quadratic equation in terms of x^2. You can solve it by factoring or using the quadratic formula.
Factoring the equation:
(x^2 - 9)(x^2 - 7) = 0
Setting each factor equal to zero:
x^2 - 9 = 0 or x^2 - 7 = 0
Solving for x in each equation:
x = ±√9 = ±3
x = ±√7
So, the points of intersection are (√9, 1), (-√9, 1), (√7, -1), and (-√7, -1).
To verify these answers, we can follow these steps for each question:
1. Find the derivative of the function p(t) to determine the rate of change at a specific point.
The derivative of p(t) can be found by using the quotient rule:
p'(t) = (t(t) - (t^2 - 2t + 1)(1))/t^2
= (t^2 - t^2 + 2t - 1)/t^2
= (2t - 1)/t^2
2. Once we have the derivative, we can substitute t = 4 to find the rate of change at 4 hours.
p'(4) = (2(4) - 1)/(4^2)
= (8 - 1)/16
= 7/16
Therefore, the rate at which the average production is changing at 4 hours is 7/16, or approximately 0.4375 (when expressed as a decimal).
For the second question:
1. Find the points of intersection between f(x) = x^2 - 8 and its reciprocal, 1/f(x).
To find the reciprocal of f(x), we can set up the equation:
1/f(x) = 1/(x^2 - 8)
Setting the two equations equal to each other:
x^2 - 8 = 1/(x^2 - 8)
To solve for x, we can multiply both sides of the equation by x^2 - 8:
(x^2 - 8)(x^2 - 8) = 1
Expanding and rearranging:
x^4 - 16x^2 + 64 = 1
x^4 - 16x^2 + 64 - 1 = 0
x^4 - 16x^2 + 63 = 0
Factoring:
(x^2 - 7)(x^2 - 9) = 0
From this factorization, we can find the points of intersection:
x^2 - 7 = 0 --> x^2 = 7 --> x = ±√7
x^2 - 9 = 0 --> x^2 = 9 --> x = ±√9
Therefore, the points where f(x) = x^2 - 8 and 1/f(x) intersect are (√7, -1), (-√7, -1), (√9, 1), and (-√9, 1).
By following these steps, you can verify the given answers.