5. For which linear function below does f(-1) = 9 and f(3) = 1?

Find the volume of the wedge with vertices at points (0,0,0), (1,0,0), (0,1,0), (0,0,1) by integrating the area of cross-section

Angelina_Jolie [31]

Answer:

V = 1/6 cubic units

Step-by-step explanation:

Applying the concept of integrals for volume calculation:

$V = \int\limits^b_a {S(x)} \, dx$ (1)

V = volume of the solid bounded by x = a and x = b

S(x) = cross section area of the solid, perpendicular to the x axis

From the figure we have that S is the area of a triangle that has base Z and height Y

Area of the triangle = $S(x)=\frac{y(x)*z(x)}{2}$ (2)

Calculation of y(x) and z(x)

We apply the equation of the point-slope line (plane xy):

Slope = $m = \frac{y_{2} - y_{1} }{x_{2} - x_{1}}$ (3)

Equation of the line = $y - y_{1} =m(x-x_{1} )$ (4)

Replacing the points (1,0) and (0,1) in (3):

$m=\frac{1-0}{0-1} =-1$

Replacing the point (1,0) and m = -1 in (4):

$y-0=(-1)(x-1)$

y(x) = -x + 1 (Line A-B) (5)

We apply the equation of the point-slope line (plane xz):

Slope = $m = \frac{z_{2} - z_{1} }{x_{2} - x_{1}}$ (6)

Equation of the line = $z - z_{1} =m(x-x_{1} )$ (7)

Replacing the points (1,0) and (0,1) in (6):

$m=\frac{1-0}{0-1} =-1$

Replacing the point (1,0) and m = -1 in (7):

$z-0=(-1)(x-1)$

z(x) = -x + 1 (Line A-C) (8)

Replacing (5) and (8) in (2)

$S(x) = \frac{(-x + 1) * (-x + 1)}{2} =\frac{(-x + 1)^{2} }{2}$ (9)

Replacing (9) in (1) and knowing that a = 0 and b = 1:

$V = \int\limits^1_0 {\frac{(-x + 1)^{2} }{2}} \, dx = \int\limits^1_0 {\frac{x^{2}-2x+1 }{2}} \, dx$

$V =\frac{1}{2} (\frac{x^{3} }{3} -2\frac{x^{2} }{2} +x)$ evaluated from x=0 to x=1

$V= \frac{1}{2} (\frac{1}{3} -1 +1) = \frac{1}{6}$

3 0

3 years ago

Please help! This is really confusing.

ioda

39 meters in 3 minutes means 13 meters each minute and 26 meters in two minutes. Multiply by 1,000 to convert it into 26000 millimeters.

8 0

3 years ago

Read 2 more answers

I need an answer for number 5

notka56 [123]

Answer:

Step-by-step explanation:

Go

7 0

3 years ago

A river is flowing at the rate of 2.4 miles per hour when a girl rows across it. If the girl rows at a still-water speed of 3.1

Brilliant_brown [7]

Why would you ask that question if you already know the answer?

3 0

2 years ago

Read 2 more answers

Can you please explain to me which functions are odd?

Lemur [1.5K]

A function is odd if f(-x)=-f(x).
So given a function, to check whether it is odd or not, we calculate f(-x). If it is equal to -f(x), then the function is odd; if not, it is not odd.

$\displaystyle{f(x)= -\frac{1}{2}x^4+5$

$\displaystyle{f(-x)= -\frac{1}{2}(-x)^4+5= -\frac{1}{2}x^4+5\neq-f(x)$

thus the function is not odd

$\displaystyle{f(x)=-8x^3+5x$

$\displaystyle{f(-x)=-8(-x)^3+5(-x)=8x^3-5x=-(-8x^3+5x)=-f(x)$

thus the function is odd

$\displaystyle{f(x)=- \frac{4}{x^3}-x+1$

$\displaystyle{f(-x)=- \frac{4}{(-x)^3}-(-x)+1= \frac{4}{x^3}+x+1\neq-f(x)$

thus the function is not odd

$\displaystyle{f(x)= \frac{x^5}{x^4-1}$

$\displaystyle{f(-x)= \frac{(-x)^5}{(-x)^4-1}= \frac{-x^5}{x^4-1}=- \frac{x^5}{x^4-1}=-f(x)$

thus the function is odd

$\displaystyle{f(x)=-\sqrt{2x}$

$\displaystyle{f(-x)=-\sqrt{2(-x)}= -\sqrt{-2x}$

In this particular case f(x) and f(-x) can both exist only for x=0 (because one of them is certainly negative). Thus the function is not odd

$\displaystyle{f(x)=3 \sqrt{x} -x^3$

similarly to the previous case, the Domain of f is [0, infinity) and f(-x) cannot be calculated except for x=0. So the function is not odd.

5 0

3 years ago