A rock is thrown in the air from the edge of a seaside cliff. Its height in feet is represented by f(x) = –16(x^2 – 8x – 9), where x is the number of seconds since the rock was thrown. The height of the rock is 0 feet when it hits the water.How long does it take the rock to hit the water?
Mathematics · High School · Mon Jan 18 2021
Answered on
Given the statement:
A rock is thrown in the air from the edge of a seaside cliff. Its height in feet is represented by f(x) = –16(x^2 – 8x – 9), where x is the number of seconds since the rock was thrown. The height of the rock is 0 feet when it hits the water
Determine how long does it take for the rock to hit the water.
Solution:
In order to determine the time, when the rock hits the water, we simply needed to look at the quadratic equation, ignoring first the value of -16, since it will still later scale. Think of a number, that when added, the sum is -8, whereas when multiplied the product is -9. In our case, we'll have the factor.
( x + 1) ( x - 9 )
In order to determine if the factor is correct, we simply use the FOIL Method. Multiply the first term of the first equation, to the first and last term of the second equation. Then, multiply the last term of the first equation, to the first and last term of the second equation.
To clearly see how it works, here's a step by step solution.
= (x)(x)
=x^2
First term of the first equation multiplied to the first term of the second equation.
=(x)(-9)
= -9x
First term of the first equation multiplied to the last term of the second equation.
=(x)(1)
= x
Last term of the first equation multiplied to the first term of the second equation.
=(-9)(1)
= -9
Last term of the first equation multiplied to the fast term of the second equation.
= x^2 -9x + x - 9
= x^2 - 8x - 9
Now that we've verified that our factor are correct, we equate each factor to 0, then solve for x.
x - 9 = 0
x + 1 = 0
Transpose -9 and 1 to the opposite sides of the equation, hence it must be taken to note that in tranpsoing a number, the sign changes.
x = 9
x = -1
Since we cannot have a negative time, therefore, our time will be 9 seconds.
Final answer:
= 9 seconds