1. Describe how you could approximately find the total number of particles in a rectangular box that is 2m x 2m x 3m by making no more than 12 samples.
2. You know the following facts about a function f(x): The derivative of f(x) is 6x + 4. The integral of f(x) from 1 to 3 is 40. Find f(5).
1. I have no clue what you mean by "particle" or what the question asks for.
#2 is straighforward
since f'(x) = 6x+4
f(x) = 3x^2 + 4x + c
⌠(3x^2 + 4x + c)dx from 1 to 3 = 40
⌡
= x^3 + 2x^2 + cx│ from 1 to 3 = 40
so
(27+18+3c)-(1+2+c) = 40
.
.
c=-1
then f(x)=3x^2 + 4x - 1
and f(5) = 94
1. To approximately find the total number of particles in a rectangular box, you can use the method of sampling. Here's how you can do it using no more than 12 samples for a box that is 2m x 2m x 3m:
1. Divide the length, width, and height of the box into smaller divisions or cuboids, such as cubes or rectangular prisms. For example, you can divide each dimension into 6 equal divisions to obtain 36 smaller cuboids.
2. Randomly select a sample from each smaller division. For instance, in each of the 36 smaller cuboids, randomly choose a point within it.
3. Measure or estimate the number of particles within each sample. You can count the number of particles or estimate it based on the density of particles within that specific region. Let's assume you estimate the number of particles in each sample.
4. Calculate the average number of particles for each sample by summing up the number of particles in all the samples and dividing by the total number of samples.
5. Multiply the average number of particles in each sample by the total number of smaller divisions or cuboids. In this case, with 36 smaller cuboids, multiply the average number of particles by 36. This will give you an estimation of the total number of particles in the rectangular box.
By following this process, you can approximate the total number of particles in the rectangular box, using no more than 12 samples.
2. To find f(5) based on the given information, you can use the Fundamental Theorem of Calculus which states that the derivative and integral of a function are related. Here's how you can do it:
1. Given that the derivative of f(x) is 6x + 4, integrate this expression to obtain f(x). The integral of 6x + 4 with respect to x is (3x^2 + 4x) + C, where C is the constant of integration.
2. Use the given information that the integral of f(x) from 1 to 3 is 40. This means that ∫[1 to 3] (6x + 4) dx = 40.
3. Evaluate the integral of f(x) from 1 to 3 using the antiderivative obtained in step 1 and set it equal to 40. Solve the equation for the constant of integration C.
4. Once you have determined the value of C, substitute x = 5 into the expression for f(x) obtained in step 1 and evaluate it. This will give you the value of f(5).
By following these steps, you can find the value of f(5) based on the given information about the derivative and integral of the function f(x).