Hello

Find the numbers b such that the average value of f(x) = 7 + 10x − 9x2 on the interval [0, b] is equal to 8.

barxatty [35]

Answer:

The numbers $b$ such that the average value of $f(x) = 7 +10\cdot x - 9\cdot x^{2}$ on the interval [0, b] is equal to 8 are $b_{1} \approx 1.434$ and $b_{2} \approx 0.232$ .

Step-by-step explanation:

The mean value of function within a given interval is given by the following integral:

$\bar f = \frac{1}{b-a}\cdot \int\limits^b_a {f(x)} \, dx$

If $f(x) = 7 +10\cdot x - 9\cdot x^{2}$ , $a = 0$ , $b = b$ and $\bar f = 8$ , then:

$\frac{1}{b}\cdot \int\limits^b_0 {7+10\cdot x -9\cdot x^{2}} \, dx = 8$

$\frac{7}{b}\int\limits^b_0 \, dx + \frac{10}{b} \int\limits^b_0 {x}\, dx - \frac{9}{b} \int\limits^b_0 {x^{2}}\, dx = 8$

$\left(\frac{7}{b} \right)\cdot b + \left(\frac{10}{b} \right)\cdot \left(\frac{b^{2}}{2} \right)-\left(\frac{9}{b} \right)\cdot \left(\frac{b^{3}}{3} \right) = 8$

$7 + 5\cdot b - 3\cdot b^{2} = 8$

$3\cdot b^{2}-5\cdot b +1 = 0$

The roots of this polynomial are determined by the Quadratic Formula:

$b_{1} \approx 1.434$ and $b_{2} \approx 0.232$ .

The numbers $b$ such that the average value of $f(x) = 7 +10\cdot x - 9\cdot x^{2}$ on the interval [0, b] is equal to 8 are $b_{1} \approx 1.434$ and $b_{2} \approx 0.232$ .

7 0

3 years ago

0.037 in scientific notation

Sedaia [141]

$\huge\boxed{3.7\ \times\ 10^{-2}}$

First, you need to find where you can move the decimal point to so that there is only one non-zero digit to the left of it.

Move the decimal point, and that is the first part done.

Add " $\times\ 10$ ".

For the last part - the power, you need to find how many decimal places you moved the decimal point. In this case, we moved it to the right 2 places, so the power is $^{-2}$ .

If, for example, we had moved the decimal point three places to the left, the power would be $^{3}$ .

7 0

3 years ago

In ten years, the population of Detroit fell from 950,000 to about 712,500. Find the percent change.

sergij07 [2.7K]

Answer:

25% decrease

Step-by-step explanation:

Percent change can be found by doing difference/original

(950,000 - 712,500) / 950,000

= 0.25

There was a 25% decrease in the population

5 0

3 years ago

Jacoby made and sold x candy bars in one week. His profit, P, in dollars is calculated using the formula P = R-C

Mashutka [201]

Answer:ion even know cus i just want the points

Step-by-step explanation:

5 0

3 years ago

A cubed-shape container has an edge length of 8 cm. The container is filled with decorative crystal cubes. Each cube has an edge

Leviafan [203]

The container is a cube with side length of 8 cm. If the crystals are also cubes with side length of 1 cm, then 8 crystals will fit along one side of the container. Since all the sides are the same length on a cube, we can take the 8 crystals and cube it to find the total amount of crystals in the container.

$8^3$ = 8·8·8 = 512

Each crystal costs 5 cents, so multiply 512 crystals by $0.05.

0.05(512) = $25.60

5 0

2 years ago