Write an equation that expresses the following relationship
d varies jointly with the square of w and the cube of p. In your equation, use k as the constant of proportionality
d = k * w^2 * p^3
Consider a triangle ABC. Suppose that a= 34,b=19,c=22 solve the triangle. Carry your intermediate computations to 4 decimal places
To solve the triangle, we need to find the measures of the remaining angles and sides.
First, we can use the Law of Cosines to find angle A:
cos(A) = (b^2 + c^2 - a^2) / (2 * b * c)
cos(A) = (19^2 + 22^2 - 34^2) / (2 * 19 * 22)
cos(A) = (361 + 484 - 1156) / (418)
cos(A) = -0.9392
A = acos(-0.9392)
A ≈ 152.9855 degrees
Next, we can use the Law of Sines to find angle B:
sin(B) / b = sin(A) / a
sin(B) = (sin(A) / a) * b
sin(B) = (sin(152.9855) / 34) * 19
sin(B) ≈ 0.8959
B = asin(0.8959)
B ≈ 64.8257 degrees
Finally, we can find angle C by subtracting angles A and B from 180 degrees:
C = 180 - A - B
C = 180 - 152.9855 - 64.8257
C ≈ -37.8112 degrees
Now, to find the lengths of the remaining sides, we can use the Law of Sines again:
c / sin(C) = b / sin(B)
c = (sin(C) / sin(B)) * b
c = (sin(-37.8112) / sin(64.8257)) * 19
c ≈ -11.1803
Since we cannot have a negative length for a side, it seems that there is an error in the given measurements. Triangle ABC cannot be solved with the given side lengths.
A circle has a radius of 17cm. Find the radian measure of the central angle theta that intercepts an arc of length 4cm
The radian measure of an angle is defined as the ratio of the length of the intercepted arc to the radius of the circle.
We are given that the radius of the circle is 17 cm and the length of the intercepted arc is 4 cm.
To find the radian measure of the central angle, we can use the formula:
θ = arc length / radius
θ = 4 cm / 17 cm
θ ≈ 0.2353 radians
Therefore, the radian measure of the central angle theta that intercepts an arc of length 4 cm is approximately 0.2353 radians.
The one to one function g and h are defined as follows
g=((-8,8),(-3,-7),(3,9),(5,-3))
h(x)=3x+4
Find ;
1)g^-1(-3)
2)h^-1(x)
3)(h of h^-1)(-2)
1) To find g^⁻1(-3), we need to find the input value from the original function g that corresponds to an output value of -3.
Looking at the function g, we see that there is a point (5, -3). This means that when x = 5, g(x) = -3. Therefore, g^⁻1(-3) = 5.
2) To find h^⁻1(x), we need to find the input value from the original function h that corresponds to an output value of x.
The function h(x) = 3x + 4. To find h^⁻1(x), we need to solve for x in terms of h^⁻1.
x = (h - 4) / 3
Therefore, h^⁻1(x) = (x - 4) / 3.
3) (h o h^⁻1)(-2) means plugging the output of h^⁻1 that equals -2 into the function h.
Using the result from part 2, h^⁻1(x) = (x - 4) / 3, we substitute x = -2 into h:
h(-2) = 3(-2) + 4
h(-2) = -6 + 4
h(-2) = -2
Therefore, (h o h^⁻1)(-2) = -2.
Consider a triangle ABC .suppose that B=75degrees, C=54degrees and a=61 solve the triangle
To solve the triangle, we can start by finding angle A.
Angle A = 180 - (B + C)
Angle A = 180 - (75 + 54)
Angle A = 180 - 129
Angle A = 51 degrees
Now, we can use the Law of Sines to find the length of the remaining sides.
sin(A) / a = sin(B) / b
sin(51) / 61 = sin(75) / b
b = (61 * sin(75)) / sin(51)
b ≈ 72.806 cm
sin(A) / a = sin(C) / c
sin(51) / 61 = sin(54) / c
c = (61 * sin(54)) / sin(51)
c ≈ 68.698 cm
Therefore, the lengths of the sides are approximately:
Side a ≈ 61 cm
Side b ≈ 72.806 cm
Side c ≈ 68.698 cm
Note: The lengths of the sides are rounded to three decimal places.