The pta created a cross-country trail for the meet.
a. the pta placed a trail marker in the ground every four hundred yards. every nine hundred yards, the pta set up a water station. what is the shortest distance a runner will have to run to see both a water station and trail marker at the same location?
b. there are 1,760 yards in one mile. about how many miles will a runner have to run before seeing both a water station and trail marker at the same location? calculate the answer to the nearest hundredth of a mile.
c. the pta wants to cover the wet areas of the trail with wood chips. they find that one bag of wood chips covers a 3 1/2 yard section of the trail. if there is a wet section of the trail that is approximately 50 1/4 yards long, how many bags of wood chips are needed to cover the wet section of the trail?

Hello!

a) there is a trail marker every 400 yards (there is one at 400 yards, other at 800 yards, and so on) , and a water station every 900 yards (900 yards, 1800 yards, 2700 yards). then the first trail marker and water station in the same location is in the minimal comon multiple between 400 and 900.

where MCM (a,b) = where mcd(a,b) is the maximal common denominator of a and b.

The maximal common denominator between 400 and 900 is 100 (is easy to see this, because 4 and 9 dont have any common divisor other than 1) then the minimal common multiplicator is :

M = 400*900/100 = 400*9 = 3600 yards

so the first trail yard that coincides with a water station is located at 3600 yards.

b) We alredy know that the runner needs to run 3600 yards in order to see both water station and trail marker, and now we know that 1760 yards are one mile.

Then (3600 yards/1760 yards)*1mile = 2.05 miles

then 3600 yards is equivalent to 2 miles ( rounding to the nearest integer)

c) The wet section is 50yards + 1/4 yard long, or 50.25 yards long ( because 1/4 = 0.25) each bag of wood chips cover 3 yards + 1/2 yard, or 3.5 yards.

then if you want to obtain the total number of bags needed, you need to divide the total distance that you want to cover by the distance that covers each bag; this is:

50.25/3.5 = 14.35 bags

rounding up ( because if you round down you will have less wood chips than necesary) you need 15 bags.

f(x) = - 2/3x + 5; f(5/2)

substitute 5/2 in for x:

f(5/2) = -2/3(5/2) + 5

simplify the right side:

f(5/2) = -5/3 + 5

f(5/2) = 10/3

answer:

f(5/2) = 10/3

i hope this you!