We know that a square root is a number that when multiplied by itself, makes a certain number. That sounds vague, but you probably have some intuition about this concept. You may realize that the square root of 9, for example, is 3, because 3 times 3 is 9. But -3 is also a square root of 9. Indeed, -3 times -3 is 9.
Now this is an interesting property. These two numbers both make positive 9. But what about -9? What number multiplied by itself gives -9?
You may recall that the product of two positive numbers is positive, and that the product of two negative numbers is also positive. The square root, by definition, requires that the number times itself equals its square, so we can't have two numbers of different signs. Thus, you can't "really" have the square root of a negative number.
But some clever mathematicians—Euler and Gauss—didn't accept this dogma. They defined our familiar numbers as "real" numbers, denoted ℝ, and made a new set of num
bers. These are called "imaginary" numbers (no joke), whose set of numbers are sometimes denoted ℂ (there's a bit more nuance to this definition). The imaginary numbers allow you to have a square root for a negative number. They are defined such that
, or
. This has opened up an unbelievable number of pathways in mathematics, but I'm sure you'll learn about it in due time. :)
Answer:
The volume of the geometric solid produced is 391 cubic cm ⇒ A
Step-by-step explanation:
<em>When a </em><em>right triangle is rotated about its vertical leg 360°</em><em>, then it formed </em><em>a cone</em><em> its radius is the horizontal leg of the triangle and its height is the vertical lege of the triangle.</em>
The rule of the volume of the cone is V = π r² h, where
- r is the radius of its base
- h is the length of its height
∵ Triangle XYZ is rotated 360° about the vertical side YZ
∴ It formed a cone with a radius = XZ and a height = YZ
∵
∵ YX = 6√3
∴
∵ tan(60) = √3
∴ = √3
→ By using cross multiplication
∴ 6√3 = XZ(√3)
→ Divide both sides by √3
∴ 6 = XZ
∵ XZ = r and YZ = h
∴ r = 6 and h = 6√3
→ By using the rule of the cone above
∵ V = (π) (6)² (6√3)
∴ V ≅ 391 cm³
∴ The volume of the geometric solid produced is 391 cubic cm
Given:
After taking a dose of medication, the amount of medicine remaining in a person's bloodstream, in milligrams, after x hours can be modeled by the function
To find:
Interpret the given function values and determine an appropriate domain for the function.
Solution:
The general form of an exponential function is
Where, a is the initial value, 0<b<1 is decay factor and b>1 is growth factor.
We have,
Here, 110 is the initial value and 0.83 is the decay factor.
It means, the amount of medicine in the person's bloodstream after taking the dose is 110 milligrams and the amount of medicine decreasing in the person's bloodstream with the decay factor 0.83 or decreasing at the rate of (1-0.83)=0.17=17%.
We know that an exponential function is defined for all real values of x but the time cannot be negative. So, x must be non negative.
We know that for any value of x. So, for all values of x.
Therefore, domain of the function is and the range is .
4. take 8 and divide it by 2