Jessica Zelinka, a Canadian heptathlete won a gold medal in javelin throw at the Pan American games in 2007. Suppose h(t)= -5t^2+15t+1 represents the approximate height, in meters, of a javelin t seconds after it is thrown.
a) Write a statement that corresponds to the quotient: h(t)/(t-b), where b is a positve integer.
b) Show that the statement in part a) may be written as Q(t)= (h(t)-h(b))/(t-b).
a) The statement that corresponds to the quotient h(t)/(t-b), where b is a positive integer, is: "The height of the javelin t seconds after it is thrown divided by the time elapsed since it was thrown, minus a positive integer b."
b) To prove that the statement in part a) may be written as Q(t) = (h(t) - h(b))/(t-b), we will substitute the given expression for h(t) into the quotient.
h(t) = -5t^2 + 15t + 1
Q(t) = h(t)/(t-b)
= (-5t^2 + 15t + 1)/(t-b)
Now, let's simplify Q(t):
Q(t) = (-5t^2 + 15t + 1)/(t-b)
= [(t-b)(-5t + 5b + 15) + (-5b^2 + 15b + 1)]/(t-b)
= (-5t^2 + 5tb + 15t - 15b + 1)/(t-b)
= (h(t) - h(b))/(t-b)
Therefore, the statement in part a) can indeed be written as Q(t) = (h(t) - h(b))/(t-b).
a) The statement that corresponds to the quotient h(t)/(t-b), where b is a positive integer, is "The rate of change of the javelin's height with respect to time, t seconds after it is thrown, when the javelin is at a height t-b seconds after being thrown."
b) To show that the statement in part a) may be written as Q(t)= (h(t)-h(b))/(t-b), we can substitute the given values into the expression:
Q(t) = (h(t)-h(b))/(t-b)
Q(t) = (-5t^2+15t+1 - (-5b^2+15b+1))/(t-b)
Q(t) = (-5t^2+15t+1 + 5b^2-15b-1)/(t-b)
Now we can simplify the numerator:
Q(t) = (-5t^2 + 5b^2 + 15t - 15b)/(t-b)
Q(t) = -5(t^2 - b^2) + 15(t - b)/(t-b)
Next, notice that t^2 - b^2 can be factored as the difference of squares:
Q(t) = -5(t + b)(t - b) + 15(t - b)/(t-b)
Finally, we can cancel out the common factor of (t - b) in the numerator and denominator:
Q(t) = -5(t + b) + 15
Q(t) = -5t - 5b + 15
Therefore, Q(t) = -5t - 5b + 15, which is equivalent to the expression h(t)/(t-b) and represents the rate of change of the javelin's height when it is at a height t seconds after being thrown, compared to when it is t-b seconds after being thrown.
a) The quotient h(t)/(t-b), where b is a positive integer, represents the ratio of the height of the javelin at time t to the time difference between t and b.
b) To show that h(t)/(t-b) is equivalent to (h(t)-h(b))/(t-b), we can use the fact that subtracting h(b) from h(t) does not change the ratio.
Let's start by subtracting h(b) from both the numerator and denominator of h(t)/(t-b):
[(h(t)-h(b)) / (t-b)] * [(t-b)/(t-b)]
Expanding the numerator:
[h(t)-(h(b) * (t-b))] / (t-b)
Now, we can simplify the numerator:
[h(t)-h(b)*(t-b)] = h(t)-h(b)*t + h(b)*b
Substituting the simplified numerator back into the equation:
[h(t)-h(b)*t + h(b)*b] / (t-b)
Rearranging the terms:
[-h(b)*t + h(t) + h(b)*b] / (t-b)
Now, we can see that this expression is equal to (h(t)-h(b))/(t-b), which confirms that the statement in part a) may be written as Q(t)= (h(t)-h(b))/(t-b).