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)
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23-7
= * in this step i subbed it into the equation.
= -3599 + 319
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16
It seems odd...
2) What is the end behaviour of
f(x) = 2-3x+4x^2
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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.
1) To determine the average rates of increase of height from t=7 to t=23 seconds, we need to find the change in height (h(t)) divided by the change in time (t).
First, we substitute t=7 and t=23 into the height function H(t) = -7t^2 + 5t - 11:
H(7) = -7(7)^2 + 5(7) - 11
H(23) = -7(23)^2 + 5(23) - 11
Next, we calculate the change in height:
Change in height (Δh) = H(23) - H(7)
Finally, we calculate the average rate of increase of height:
Average rate of increase = Δh / (23 - 7)
Plug in the values we found and solve for the average rate of increase.
However, it seems the calculations you provided are incorrect. Double-check your calculations to get the correct answer.
2) To determine the end behavior of a function, we need to analyze the highest-degree terms in the numerator and denominator of the function expression.
In this case, the highest-degree term in the numerator is 4x^2, and the highest-degree term in the denominator is 8x^2. As x approaches positive or negative infinity, the effect of smaller terms becomes negligible, and we can ignore them for determining the end behavior.
Therefore, the end behavior of the function f(x) = (2-3x+4x^2) / (8x^2+5x-11) is the same as the end behavior of the ratio of the highest-degree terms, which is 4x^2 / 8x^2 = 1/2.
As x approaches positive or negative infinity, the function approaches a value of 1/2.
3) To solve the equation 2^(2x) - 4(2^x) + 3 = 0 for x:
Let's make a substitution: Let y = 2^x. The equation then becomes: y^2 - 4y + 3 = 0.
Now we have a quadratic equation in terms of y. Factoring the quadratic equation, we get: (y - 1)(y - 3) = 0.
Setting each factor equal to zero, we have two possible values for y:
y - 1 = 0 or y - 3 = 0
Solving these equations, we find: y = 1 or y = 3.
Substituting y back in terms of x, we get:
2^x = 1 or 2^x = 3.
For 2^x = 1, the only solution is x = 0 since 2^0 = 1.
For 2^x = 3, we can take the logarithm of both sides to solve for x. Taking the logarithm with base 2, we get: x = log2(3).
So the solutions to the original equation 2^(2x) - 4(2^x) + 3 = 0 are x = 0 and x = log2(3).