Scenario 1) You and your friend enjoy riding your bicycles. Today is a beautiful sunny day, so the two of you are taking a long ride out in the country side. Leaving your home in Sunshine, you ride 6 miles due west to the town of Happyville, where you turn south and ride 8 miles to the town of Crimson. When the sun begins to go down, you decide that it is time to start for home. There is a road that goes directly from Crimson back to Sunshine. If you want to take the shortest route home, do you take this new road, or do you go back the way you came? Justify your decision. How much further would the longer route be than the shorter route? Assume all roads are straight.
Scenario 2) A newly-planted tree needs to be staked with three wires. Each wire is attached to the trunk 3 ft. above the ground, and then anchored to the ground 4 ft. from the base of the tree. How much wire is needed for 6 trees?
Scenario 3) Jill’s front door is 42” wide and 84” tall. She purchased a circular table that is 96 inches in diameter. Will the table fit through the front door? Explain using approximations.
(1)hypotenuse = straight home
sqrt(36+64) = 10
straight is 10 miles
same way is 6+8 = 14 miles
(2) hypotenuse of 3,4 5 right triangle is 5
5*6 = 30
plus enough for the knots :)
(3)diagonal (hypotenuse) length is the longest
sqrt (42^2 + 84^2) = 93.9, so no
Scenario 1: Well, if you're looking for the shortest route home, it's time to take a detour through clown town, my friend! You see, if you take the new road directly from Crimson back to Sunshine, it would actually be shorter than going back the way you came. Going back the way you came would mean going 6 miles west and then 8 miles north, which adds up to 14 miles. But if you take the new road, you're just cutting directly from Crimson to Sunshine, which would be a straight shot of hypotenuse of a right triangle, my mathematician pal! So it would be like your own personal shortcut! Now, I'm no Pythagoras, but using the good ol' Pythagorean theorem, we can figure out the length of that hypotenuse. The distance west is 6 miles and the distance south is 8 miles, so the hypotenuse would be the square root of 6 squared plus 8 squared, which is 10 miles. So, going the new road route would be 4 miles shorter than going back the way you came. Smooth sailing, my friend!
Scenario 2: Oh boy, sounds like you've got a whole forest of trees to stake there! If each tree needs to be staked with three wires, and each wire is attached 3 ft. above the ground and anchored 4 ft. from the base of the tree, then we can calculate the length of wire needed for each tree. You see, it's like drawing a triangle between the base of the tree, where the wire is anchored, and the point where the wire is attached to the trunk. Using a little Pythagorean magic again, we can find the length of that wire! So, the wire attached 3 ft. above the ground and the wire anchored 4 ft. from the base of the tree form the legs of a right triangle. Using the lovely Pythagorean theorem, we can calculate the length of the hypotenuse, which is the length of wire needed. The length of the hypotenuse is the square root of 3 squared plus 4 squared, which comes out to be about 5 ft.! So, for each tree, you'll need about 5 ft. of wire. Now you just multiply that by the number of trees, which is 6, and you'll need a grand total of about 30 ft. of wire. Good luck wrangling all those wires, my friend!
Scenario 3: Ah, the age-old question of whether a circular table can fit through a door. It's like a philosophical conundrum! Now, let's get down to the approximate measurements, because approximations are as close as we'll get to precision in the world of clowns. Jill's front door is 42" wide and 84" tall. The circular table she purchased is 96 inches in diameter. Now, let's average out those dimensions because approximations are all about making things simpler! The average of 42" and 84" is 63". So, let's pretend Jill's door is 63" by 63". Now, if we're trying to fit that circular table through the door, we need to see if the diameter of the table is smaller or equal to the width of the door. So, if the diameter of the table is 96 inches, we just need to see if that's less than or equal to 63 inches. And alas, my friend, 96 is greater than 63. So, it seems like that circular table won't fit through the front door. But hey, who needs a front door when you can make the table the centerpiece of your living room? Let those walls expand their horizons!
Scenario 1) To determine whether to take the new road or go back the way you came, you need to figure out the distances for both routes.
If you take the new road directly from Crimson back to Sunshine, you would be traveling in a straight line. This can be calculated using the Pythagorean theorem. The distance from Happyville to Crimson is 8 miles, and the distance from Sunshine to Happyville is 6 miles.
Using the Pythagorean theorem, you can calculate the distance from Sunshine to Crimson:
distance^2 = 6^2 + 8^2
distance^2 = 36 + 64
distance^2 = 100
distance = sqrt(100)
distance = 10 miles
So if you take the new road, the distance traveled would be 10 miles.
To determine the distance if you go back the way you came, you would need to add the distances from Happyville to Crimson and Happyville to Sunshine:
distance = 8 + 6
distance = 14 miles
Therefore, going back the way you came would be a longer route by 4 miles (14 - 10).
Scenario 2) To figure out the amount of wire needed for 6 trees, you need to calculate the total length of wire required for one tree and then multiply that by 6.
Each wire is attached to the trunk 3 ft. above the ground and anchored 4 ft. from the base of the tree. So you can form a right-angle triangle with the tree trunk as the hypotenuse, and the other two sides being 3 ft. and 4 ft.
Using the Pythagorean theorem, you can find the length of wire needed for one tree:
length^2 = 3^2 + 4^2
length^2 = 9 + 16
length^2 = 25
length = sqrt(25)
length = 5 ft
Therefore, for one tree, you would need 5 ft of wire.
To find the total amount of wire needed for 6 trees, you can multiply the wire length for one tree by 6:
total wire needed = 5 ft * 6 trees
total wire needed = 30 ft
So, you would need 30 ft of wire for 6 trees.
Scenario 3) To determine if the circular table will fit through the front door, you need to compare the dimensions of the door and the table.
The front door is 42" wide, and the table has a diameter of 96". To determine if the table can fit through the door, you need to compare the width of the door (42") to the diameter of the table (96").
Since the diameter is greater than the width of the door, the table will not fit through the front door.
When making approximations, always choose the larger value to make sure the object can fit without any issues or complications.