Response:
Step-by-step breakdown:
When you sketch that diagram (great description, by the way!), what you essentially have is a right triangle with a base of 32 and a hypotenuse of 45. The right angle resides at one of the base's ends, and x represents the vertex angle. We must find this vertex angle first to determine the angle of depression from the second bird to the watcher. The side measuring 32 is opposite to angle x, with 45 being the hypotenuse; hence, the trigonometric relation we need is sine:
and
sin(x) =.711111111
Go to your calculator, press the 2nd key followed by the sin key, and your display will show:
then, enter in your decimal.711111111 and hit equals. You should arrive at an angle of 45.325. That angle is x. However, that's not the angle of depression. The angle of depression is the complementary angle to x.
Angle of depression = 90 - angle x and
Angle of depression = 90 - 45.325, resulting in
Angle of depression = 44.67 or 44.7 degrees.
The P-value to evaluate the claim that the mean length of pencils produced in this factory equals 18.0 cm is 0.00736. Step-by-step explanation: In this case, a quality control specialist extracted a random sample of 45 pencils from the assembly line, which exhibited a mean length of 17.9 cm. With a known population standard deviation of 0.25 cm, we denote by the population mean length for pencils produced in the factory. Thus, Null Hypothesis: = 18.0 cm (indicating that the population mean length equals 18.0 cm). Alternate Hypothesis: ≠ 18.0 cm (suggesting different from 18.0 cm). We apply the one-sample z-test since the population standard deviation is known. The test statistic yields: T.S. ~ N(0,1), with the sample mean length 17.9 cm and population standard deviation 0.25 cm for a sample size of 45. Hence, the calculated test statistic is -2.68. The corresponding P-value is derived from P(Z < -2.68) = 1 - P(Z > 2.68), equating to 1- 0.99632 = 0.00368. For a two-tailed test, the resulting P-value computes to 2 * 0.00368 = 0.00736.
To minimize the amount of paper used, the dimensions should be perfectly matched such that the diameter equals the height. Therefore, if we denote the diameter as x, then the height equals x and the radius is x/2.
Consequently, the cone's volume is expressed as:
V=1/3πr^2h
=1/3*π*(x/2)^2*x
=0.262x^3
From this, we can derive that:
33=0.262x^3
thus, we find x^3=125.95
solving gives x=(125.95)^(1/3)
resulting in x=5.0126=5.01 cm
thus, the diameter and height are both 5.01cm.
Percentage change in price = 13.80%.
