Question

1. modeling with mathematics starting from 300 feet away, a car drives toward you. it then passes by you at a speed of 48 feet per second. the distance d (in feet) of the car from you after t seconds is given by the equation d = |300 - 48t|. at what times is the car 60 feet from you? 46. making an argument your friend says that the absolute - value equation |3x + 8| - 9 = - 5 has no solution because the constant on the right - side of the equation is negative. is your friend correct? explain. 47. modeling with mathematics you randomly survey students about year - round school. the results are shown in the graph. the error given in the graph means that the actual percent could be 5% more or 5% less than the percent reported by the survey. a. write and solve an absolute value equation to find the least and greatest percents of students who could be in favor of year - round school. b. a classmate claims that 1/3 of the student body is actually in favor of year - round school. does this conflict with the survey data? explain.

Explanation:

45.

Step1: Set up the absolute - value equation

Set \(d = 60\) in the equation \(d=\vert300 - 48t\vert\), so we get \(\vert300 - 48t\vert=60\).

Step2: Consider the two cases for absolute - value

Case 1: \(300−48t = 60\).
Subtract 300 from both sides: \(-48t=60 - 300=-240\).
Divide both sides by \(-48\): \(t=\frac{-240}{-48}=5\).
Case 2: \(300−48t=-60\).
Subtract 300 from both sides: \(-48t=-60 - 300=-360\).
Divide both sides by \(-48\): \(t = \frac{-360}{-48}=7.5\).

Step1: Isolate the absolute - value term

Given \(\vert3x + 8\vert-9=-5\), add 9 to both sides of the equation.
We get \(\vert3x + 8\vert=-5 + 9=4\).

Step2: Consider the two cases for absolute - value

Case 1: \(3x+8 = 4\).
Subtract 8 from both sides: \(3x=4 - 8=-4\).
Divide both sides by 3: \(x=-\frac{4}{3}\).
Case 2: \(3x + 8=-4\).
Subtract 8 from both sides: \(3x=-4 - 8=-12\).
Divide both sides by 3: \(x=-4\).
So the friend is incorrect.

Step1: Set up the absolute - value equation

Let \(p\) be the actual percent of students in favor. The reported percent is 32% and the error is 5%.
The absolute - value equation is \(\vert p - 32\vert=5\).

Step2: Consider the two cases for absolute - value

Case 1: \(p - 32=5\).
Add 32 to both sides: \(p=5 + 32=37\).
Case 2: \(p - 32=-5\).
Add 32 to both sides: \(p=-5 + 32=27\).

Answer:

The car is 60 feet from you at \(t = 5\) seconds and \(t = 7.5\) seconds.

46.