There exist an abbreviation that ALL - S - T - C where all trigonometric functions in first quandrant are positive. S, T, and C are the first letters of the trigonometric functions that are positive in quadrant 2, 3, and 4, respectively. This also means that in the same quadrant, their reciprocals are also positive. For the given above, it is in Quadrant 3 where T is positive and cosine is negative.

5 0

3 years ago

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Semicircles

Studentka2010 [4]

Answer:

The area of the shaded portion of the figure is $9.1\ cm^2$

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

The shaded area is equal to the area of the square less the area not shaded.

There are 4 "not shaded" regions.

step 1

Find the area of square ABCD

The area of square is equal to

$A=b^2$

where

b is the length side of the square

we have

$b=4\ cm$

substitute

$A=4^2=16\ cm^2$

step 2

We can find the area of 2 "not shaded" regions by calculating the area of the square less two semi-circles (one circle):

The area of circle is equal to

$A=\pi r^{2}$

The diameter of the circle is equal to the length side of the square

$r=\frac{b}{2}=\frac{4}{2}=2\ cm$ ---> radius is half the diameter

substitute

$A=\pi (2)^{2}$

$A=4\pi\ cm^2$

Therefore, the area of 2 "not-shaded" regions is:

$A=(16-4\pi) \ cm^2$

and the area of 4 "not-shaded" regions is:

$A=2(16-4\pi)=(32-8\pi)\ cm^2$

step 3

Find the area of the shaded region

Remember that the area of the shaded region is the area of the square less 4 "not shaded" regions:

$A=16-(32-8\pi)=(8\pi-16)\ cm^2$

---> exact value

assume

$\pi =3.14$

substitute

$A=(8(3.14)-16)=9.1\ cm^2$

8 0

3 years ago

The graph shows the functions f(x), p(x), and g(x): Graph of function g of x is y is equal to 1 plus the quantity 1.5 raised to

satela [25.4K]

Part A:

Given that the <span>straight line p(x) joins the ordered pairs (0, 2) and (1, -5), thus the equation of the line joining ordered pairs (0, 2) and (1, -5) is given by

$\frac{y-2}{x} = \frac{-5-2}{1} =-7 \\ \\ \Rightarrow y-2=-7x \\ \\ \Rightarrow y=-7x+2$

Thus, p(x) = -7x + 2

</span>Given that the <span>straight line f(x) joins the ordered pairs (4, 1) and (2, -3), thus the equation of the line joining ordered pairs (4, 1) and (2, -3) is given by

$\frac{y-1}{x-4} = \frac{-3-1}{2-4} =\frac{-4}{-2}=2 \\ \\ \Rightarrow y-1=2(x-4)=2x-8 \\ \\ \Rightarrow y=2x-7$

Thus, f(x) = 2x - 7
</span>
The solution to the pair of equations represented by p(x) and f(x) is given by

p(x) = f(x)
⇒ -7x + 2 = 2x - 7
⇒ -7x - 2x = -7 - 2
⇒ -9x = -9
⇒ x = -9 / -9 = 1

Substituting for x into p(x), we have

p(1) = -7(1) + 2 = -7 + 2 = -5

Therefore, the solution to the pair of equations represented by p(x) and f(x) is (1, -5)

Part B:

From part A, we have that f(x) = 2x - 7

when x = -8

f(-8) = 2(-8) - 7 = -23

Thus, (-8, -23) is a solution to f(x).

When x = -10

f(-10) = 2(-10) - 7 = -27

Thus, (-10, -27) is a solution to f(x).

Therefore, two solutions of f(x) are (-8, -23) and (-10, -27).

Part C:

From part A, we have that p(x) = -7x + 2, given that $g(x) = 1 + 1.5^x$

From the graphs of p(x) and g(x), we can see that the two graphs intersected at the point (0, 2).

Therefore, the solution to the equation p(x) = g(x) is (0, 2).

3 0

3 years ago