What is the exponential function that best models the number of gnats the scientists have gathered after the number of hours lis
ted in the table below?
1 answer:
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Answer:
Step-by-step explanation:
Notice that there is a right angle triangle formed with sides as follows:
hypotenuse is the actual 1500 ft string. The acute angle 43 degrees is opposite to the segment that represents the height of the kite from the ground. Therefore, the trigonometric ratio that we can use to find that opposite side to the given angle, is the sine function as shown below:
Answer:
Step-by-step explanation:
For a. we start by dividing both sides by 200:
In order to solve for x, we have to get it out from its position of an exponent. Do that by taking the natural log of both sides:
Applying the power rule for logs lets us now bring down the x in front of the ln:
x * ln(1.05) = ln(1.885)
Now we can divide both sides by ln(1.05) to solve for x:
Do this on your calculator to find that
x = 12.99294297
For b. we will first apply the rule for "undoing" the addition of logs by multipllying:
Simplifying gives you
Applying the power rule allows us to bring down the 3 in front of the ln:
3 * ln(x) = 5
Now we can divide both sides by 3 to get
Take the inverse ln by raising each side to e:
The "e" and the ln on the left undo each other, leaving you with just x; and raising e to the power or 5/3 gives you that
x = 5.29449005
For c. begin by dividing both sides by 20 to get:
"Undo" that e by taking the ln of both sides:
When the ln and the e undo each other on the right you're left with just .1x; on the left we have, from our calculators:
-.6931471806 = .1x
x = -6.931471806
Question d. is a bit more complicated than the others. Begin by turning the base of 4 into a base of 2 so they are "like" in a sense:
Now we will bring over the -8 by adding:
We can turn this into a quadratic of sorts and factor it, but we have to use a u substitution. Let's let
When we do that, we can rewrite the polynomial as
This factors very nicely into u = 4 and u = 2
But don't forget the substitution that we made earlier to make this easy to factor. Now we have to put it back in:
For the first solution, we will change the base of 4 into a 2 again like we did in the beginning:
Now that the bases are the same, we can say that
x = 2
For the second solution, we will raise the 2 on the right to a power of 1 to get:
Now that the bases are the same, we can say that
x = 1