Question question 1.
the speed of sound is 760 mph in air. a gun is fired 1/2 mile from you. how long will it take for the sound to reach you?
question 2.
a football field is a rectangle. the length of the field from goal line to goal line is 46 yards more than the width. the perimeter of the field is 308 yards. what are the dimensions of the field?
question 3.
you have 5 gallons of a solution which is 20% alcohol (and 80% water). you need to add enough 100% alcohol to make the solution 50% alcohol. how many gallons of 100% alcohol will you add?
question 4.
a baseball diamond is a square. the distance from base to base is 90 feet. what is the distances from home plate to second base?
question 5.
solve the inequality !3x^2 - 7x + 2! greater than 8 write your answer in interval form.
question 6.
you have 100 feet of fence to build a rectangular pen to house animals. a government rule requires that your pen encloses at least 600 square feet. what is the interval of choices for the length of the pen that will meet the standards?
Answer question 1.
the speed of sound is 760 mph in air. a gun is fired 1/2 mile from you. how long will it take for the sound to reach you?
Since the 1/2 miles is in miles, and we want the units to cancel,
you need to multiply by hour/mile. That is 1/760. (1/2)(1/760) = (1/1520) hours.
Multiply by 60 min/hour and 60 sec/min, and see the units cancels to give seconds.
That is, take 60*60/1520 = 2.37 sec.
question 2.
a football field is a rectangle. the length of the field from goal line to goal line is 46 yards more than the width. the perimeter of the field is 308 yards. what are the dimensions of the field?
The width is 46 yards, so lets say W = 46.
The perimeter is 2W + 2L, so 2W + 2L = 308.
Dividing by 2 gives W + L = 154.
So if W = 46, we have 46 + L = 154, so L = 154 - 46 = 108.
This means the dimensions are 108 yards long by 46 yards wide.
question 3.
you have 5 gallons of a solution which is 20% alcohol (and 80% water). you need to add enough 100% alcohol to make the solution 50% alcohol. how many gallons of 100% alcohol will you add?
You have 5 gallons which is 20% alcohol, so that makes 1 gallon of alcohol and 4 gallons of water. The mixture is suppose to be 50% alcohol, so that means there should be the same number of gallons of each. Since there are 4 gallons of water and only 1 gallon of alcohol, we need 3 more gallons of alcohol added. That would make 4 gallons of alcohol and 4 gallons of water.
question 4.
a baseball diamond is a square. the distance from base to base is 90 feet. what is the distances from home plate to second base?
The distance from base to base is the hypoteneuse of a triangle with sides that are both 90.
This distance is then 90√2. Since the diamond is in the middle, half that is the distance to home and half that is the distance to 2nd plate. Half of 90√2 is 45√2.
question 5.
solve the inequality !3x^2 - 7x + 2! greater than 8 write your answer in interval form.
For |3x² - 7x + 2| > 8, we need to know where the function 3x² - 7x + 2 is positive and
where the function is negative. Note that it can be factored into (3x-1)(x-2). This means that the zeroes are at x=1/3 and x=2. In between these two values, 3x-1 is positive and x-2 is negatvie. Since they are multiplied together, that makes the function negative in that interval.
Outside that interval, they are both positive on the right and both negative on the left.
Whether two positives or two negatives are multiplied together, a positive is gotten.
What we need to do, then, is solve 3x² - 7x + 2 > 8 for x<1/3 or x>2
and we need to solve -3x² + 7x - 2 => 8 in the middle.
For the 1st one, subtract 8 from both sides. This gives 3x² - 7x - 6 > 0.
The quadratic formula tells us that x = (7±√(49+72))/(2*3) = (7±√121)/6 = (7±11)/6.
That is, x=-2/3 or x=3. Note there is a 3 in front of the x², so we take 3(x+2/3)(x-3),
and this gives us back the function, so that's good.
Since the coefficient in front of the x² is positive,
this will be an upwards sloping parabola on both ends.
This means it will be positive for x<-2/3 and x>3.
Note that both of these are in the range for the first one, so that is the answer.
For the 2nd one, we subtract 8 and see where the function changes sign.
Once this is done, I will compare it to the range of 1/3 < x < 2.
The function is -3x² + 7x - 10 = 0. The factors are (x+1)(-3x+10).
The function is positive between -1 and 10/3.
Since we had x between 1/3 and 2, this means that it is true everywhere in that interval
since -1 is less than 1/3 and 10/3 = 3 1/3 > 3.
So this applies for all x that satisfy 1/3 < x < 2.
question 6.
you have 100 feet of fence to build a rectangular pen to house animals. a government rule requires that your pen encloses at least 600 square feet. what is the interval of choices for the length of the pen that will meet the standards?
Let W be the width and L be the length. The area is WL.
It is given that we have 100 feet, and we know that 2W + 2L = 100, or W + L = 50.
We want WL to be at least 600. Note that L = 50 - W, so we have and area of W(50-W).
Setting that to 600 and solving gives 50W - W² = 600, or 0 = W² - 50W + 600.
This factors into (W-20)(W-30). We want this value to be less than 0 (since this is 600 - area),
so we need W between 20 and 30.
