1) A helium balloon is rising according to the function H(t) = -7x^2 + 5x - 11 where h(t) is the height in meters after t seconds. Determine the average rates of increase of height from t=7 to t=23 seconds.
I tried
h(23) - h(7)
------------
23-7
= * in this step i subbed it into the equation.
= -3599 + 319
-------------
16
It seems odd...
2) What is the end behaviour of
f(x) = 2-3x+4x^2
-----------
8x^2+5x-11
I understand end behaviour..but can someone help rearrange the first equation?
3) 2^2x - 4 (2^x) + 3 = 0
Solve for x.
This one I don't get at all.m
It is odd all right. First of all if it is h(t), why is it written with x and no t on the right? The whole thing is a typo.
The -x^2 term means it will sink, not rise, as x gets large. That is why your average speed comes out negative.
2-3x+4x^2
-----------
8x^2+5x-11
4 x^2 -3 x + 2
------------------
8 x^2 + 5 x -11
as x gets big, the x^2 terms are much bigger than the terms in x and the constants so
4 x^2
------ = 1/2
8 x^2
2^2x - 4 (2^x) + 3 = 0
The trick is to see that:
2^2x = (2^x)^2
so say y = 2^x
and we have
y^2 - 4 y +3 = 0
(y-3)(y-1)=0
y = 3 or y = 1
so 2^x = 3 or 2^x = 1
x log 2 = log 3 or x log 2 = log 1
x =.477/.301 or x = 0
1) To determine the average rates of increase of height from t=7 to t=23 seconds, we can use the formula:
Average rate of increase = (change in height) / (change in time)
In this case, the change in time is 23 - 7 = 16 seconds.
To find the change in height, we need to evaluate h(23) and h(7) using the given function:
h(23) = -7(23)^2 + 5(23) - 11
h(7) = -7(7)^2 + 5(7) - 11
Now, we can substitute these values into the formula for the average rate of increase:
Average rate of increase = (h(23) - h(7)) / (23 - 7)
After calculating the numerator and the denominator, you should simplify to get the final result.
Regarding your calculation:
You correctly substituted the values of t into the function, but there seems to be an error in the arithmetic calculations. To correct it, double-check the calculations for h(23) and h(7), and then re-evaluate the average rate of increase.
2) The end behavior of a function describes what happens to the function's values as x approaches positive or negative infinity. To determine the end behavior of the given function:
f(x) = (2-3x+4x^2) / (8x^2+5x-11)
First, we need to analyze the highest power terms in the numerator and denominator. In this case, the highest power term is x^2.
For x^2, the leading coefficients are 4 in the numerator and 8 in the denominator.
When x approaches positive or negative infinity, the terms involving x^2 dominate the function. Since both the numerator and denominator have positive coefficients for x^2, we can determine that the end behavior will be:
- As x approaches positive infinity, f(x) approaches positive infinity.
- As x approaches negative infinity, f(x) approaches positive infinity.
Therefore, the end behavior of the function f(x) is that it approaches positive infinity as x approaches either positive or negative infinity.
3) To solve the equation 2^(2x) - 4(2^x) + 3 = 0 for x, we can use a substitution to create a quadratic equation.
Let's substitute 2^x = u. Then the equation becomes:
u^2 - 4u + 3 = 0
Now, we can factor this quadratic equation:
(u - 3)(u - 1) = 0
Setting each factor equal to zero:
u - 3 = 0 --> u = 3
u - 1 = 0 --> u = 1
Since we originally substituted u = 2^x, we need to solve for x:
2^x = 3 --> x = log2(3)
2^x = 1 --> x = log2(1)
The equation 2^(2x) - 4(2^x) + 3 = 0 has two solutions for x: x = log2(3) and x = log2(1).
Please note that log2 denotes the logarithm base 2.