h(t) = -16t^2 + b. h(0) = 3364. Approximately how many seconds after being dropped does the function estimate the object will hit the ground? A) 7.25 B) 14.50 C) 105.13 D) 210.25

Question

SAT Practice Test #7 Module 2 Question 19

h(t) = −16t^2 + b. The function h estimates an object’s height, in feet, above the ground t seconds after the object is dropped, where b is a constant. The function estimates that the object is 3,364 feet above the ground when it is dropped at t = 0. Approximately how many seconds after being dropped does the function estimate the object will hit the ground? A) 7.25 B) 14.50 C) 105.13 D) 210.25

Answer 1

Knowing that the function f(t)=-16t^2+b tell us that this is a parabola that has not been shifted off the y-axis (no "t" term) and b is the y-intercept, Since f(0) is the y-intercept, the function is f(t)=-16t^2+3364.

This is an SAT question, so you have Desmos available on the test. Change the t to x and plug it into Desmos. Find the positive x-intercept, because that is where the object hits the ground. That intercept is 14.5, so the answer is B..

Answer 2

We're given that h(t) = -16t² +b and h(0)=3364

So we know that when t=0 the height is 3364 ft. We can substitute these two values into the function h(t) to find the value of the constant "b".

We get 3364 = -16(0)² +b

3364 = 0 + b

Therefore b = 3364.

So now the function is h(t) = -16t² + 3364.

When the object hits the ground h(t) =0

So substituting h(t)=0 into the function, we get:

0 = -16t² + 3364

16t² = 3364

t² = 3364/16

= 841/4

t = square root of 841/4

= 29/2

t = 14.50

The answer is B.

Answer 3

Given:

h(t) = -16t² + b

h(0) = 3364

Observations:

On SAT, always gather some bits of evidence from the description early, and decide which may be relevant. Some may not be.

(a) When 0 is plugged into h(t) for t = 0, we will be left with only ‘b’. Relevant. File that.

(b) h(t) describes an inverted parabola in time ’t' at Vertex (0, b). Not Relevant yet. Maybe.

(c) The problem description does not state the units of h(t), and we are _not_ asked for units among the answers.

We file this as a mental note, as we proceed: Perhaps we quickly observe that given that the usual “free fall” equations may either involve acceleration due to Earth’s gravity as g = 32 ft/sec² or g = 9.81 m/sec², and the usual free fall equations involve the term (1/2)gt² , we notice g/2 = 16 makes it easy to conclude that h(t) is in feet, but just as easy to dismiss this as irrelevant. Not Relevant. Move on.

(d) Problem description does not specify “when” nor from "what height” the object will be dropped.

We may need to assume something at some point, or find another hint as we go. Unknown. Relevant.

Solution:

h(0) = -16(0)² + b ==> h(0) = b, also given that h(0) = 3364 ==> b = 3364

So far we have, using observation (a): h(t) = -16t² + 3364

We still do not know what to make of observation (d), we have found no new information so far.

So we assume that object will be dropped from the highest point, which is what ? Remember that irrelevant ‘inverted parabola vertex’ mental note ?

Given h(t) = -16t² + 3364, 3364 is the maximum because the term -16t² will always be negative and cause h(t) < 3364 for all t ≠ 0.

We need to assume object drop at Maximum height. We are not told otherwise, we are not told that this is so either.

We want to know specific duration ’T' of the fall, when the object hits the ground at h(T) = 0 ==>

-16T² + 3364 = 0 ==> 16T² = 3364 ==> T² ⁼ 3364 / 16 ==>

T = ±√(3364 / 16) = ±√(1682 / 8) = ±√(841 / 4) = ±(29 / 2) = ± 14.5

# Note on Mental Math during SAT: √841 is close to √900 = 30. Which numbers not far from, and less than 30, when squared, end with the digit 1 of 841 ? 9x9 = 81, candidate, so maybe 29. 8x8 ends in 4, 7x7 ends in 9 <keep going>, we have memorized that 25x25 = 625 (right?), too far back from 841, so we go with 29.

Quick mental Algebra check: (30 - 1)(30 - 1) = 30² - 2(30) + 1… yes.

This is SAT after all, where reasonable numbers are often selected for a reason to obviate the need for Calculator!

We discard the negative answer, choose + 14.5 for elapsed free fall time since t = 0

Answer: (B) 14.50

Answer 4

First find "b" in the function h(t) = -16t² + b by using the information provided h(0) = 3364.

3364 = -16(0)² + b

3364 = b

Therefore the complete function is h(t) = -16t² + 3364

Now, set h(t) = 0 since the height at ground level is 0 feet.

0 = -16t² + 3364

16t² = 3364

t² = 3364/16

t² = 210.25

t = √210.25

t = 14.5 seconds

Answer 5