Question

An object dropped from a height falls a distance of \(d\) feet in \(t\) seconds. The formula that describes this relationship is \(d=16 t^{2} .\) A math book is dropped from the top of a building that is 400 feet high.\a. How far has the book fallen in \(1 \frac{1}{2}\) seconds? b. After \(3 \frac{1}{2}\) seconds, how far above the ground is the book? c. Another book was dropped from the top of a building across the street. It took \(6 \frac{1}{2}\) seconds to hit the ground. How tall is that building?

Short answer

Answer: After 1.5 seconds, the book has fallen 36 feet. After 3.5 seconds, the book is 204 feet above the ground. The height of the other building is 676 feet.

Step-by-step solution

Part a: Finding distance fallen in 1.5 seconds

First, convert the mixed number \(1\frac{1}{2}\) to an improper fraction, which would be \(\frac{3}{2}\) or simply 1.5. Then, plug it into the formula \(d = 16t^2\). So, \(d = 16(1.5)^2\). Calculate the value to find the distance fallen.

Calculation of distance fallen in 1.5 seconds

Compute the distance as follows: \(d = 16(1.5)^2 = 16(2.25) = 36\) So, after \(1\frac{1}{2}\) seconds, the book has fallen 36 feet.

Part b: Finding distance above ground after 3.5 seconds

Similar to part a, first convert \(3\frac{1}{2}\) into an improper fraction or decimal, which is \(\frac{7}{2}\) or 3.5. Then, plug it into the formula \(d = 16t^2\). Calculate the distance fallen after 3.5 seconds. To find the distance above the ground, subtract the fallen distance from the total height of the building (400 feet).

Calculation of distance above ground after 3.5 seconds

Compute the distance fallen after 3.5 seconds: \(d = 16(3.5)^2 = 16(12.25) = 196\) Now, subtract the fallen distance from the total height: \(400 - 196 = 204\) After 3.5 seconds, the book is 204 feet above the ground.

Part c: Finding the height of another building

Another book took \(6\frac{1}{2}\) seconds (6.5 seconds) to hit the ground on another building. Utilize the formula \(d = 16t^2\) and plug in the value of time (6.5 seconds) to calculate the height of this building.

Calculation of the height of another building

Compute the height (distance from top to bottom) of the other building: \(d = 16(6.5)^2 = 16(42.25) = 676\) So, the height of the building across the street is 676 feet.