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 km due west to the town of Happyville, where you turn south and ride 8 km 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 m. above the ground, and then anchored to the ground 4 m. from the base of the tree. How much wire is needed for 6 trees?
Scenario 3: Jill’s front door is 98 cm wide and 190cm tall. She purchased a circular table that is 96 cm in diameter. Will the table fit through the front door? Determine the radius of the largest table that will fit through the door? Explain using mathematics.
Scenario 1:
To determine whether you should take the new road or go back the way you came, you need to find which route is shorter.
To do this, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, we can consider the distance you rode from Crimson to the new road as the hypotenuse of a right-angled triangle, and the distances you rode from Sunshine to Happyville and from Happyville to Crimson as the other two sides.
Using the Pythagorean theorem, we can calculate the lengths of these sides. Let's call the distance from Sunshine to Happyville "a" and the distance from Happyville to Crimson "b". Then, the distance from Crimson to the new road, which we'll call "c", can be found using the equation:
c^2 = a^2 + b^2
The shorter route will be the one with the smaller value of c. Calculate the values of a, b, and c and compare them to determine which route is shorter.
Scenario 2:
To calculate the amount of wire needed for each tree, we need to find the perimeter of the triangular area formed by the wire and the ground.
The wire is attached to the trunk of the tree 3 m above the ground and anchored at a point 4 m from the base of the tree. This creates a right-angled triangle with one side measuring 3 m, another side measuring 4 m, and the wire as the hypotenuse.
Using the Pythagorean theorem, we can find the length of the wire. Let's call it "w". The equation will be:
w^2 = 3^2 + 4^2
Once we find the length of the wire, we can multiply it by the number of trees (6) to get the total amount of wire needed for all the trees.
Scenario 3:
To determine if the circular table will fit through Jill's front door, we need to compare the diameter of the table to the width and height of the door.
The diameter of the table is 96 cm, so its radius will be half of that, which is 48 cm.
To determine if the table will fit through the door, we need to compare the diameter of the table with the smaller dimension of the door, which is the width (98 cm). The table will fit through the door if its diameter is smaller than or equal to the width.
In this case, the diameter (96 cm) is smaller than the width (98 cm), so the table will fit through the door.
To determine the largest table that will fit through the door, we need to find the radius of a table where the diameter is equal to the width of the door (98 cm).
Since the diameter is twice the radius, we can calculate the radius by dividing the width by 2:
Radius = 98 cm / 2 = 49 cm
Therefore, the largest table that will fit through the door has a radius of 49 cm.
#1. You have gone two sides of a right triangle.
The 3rd side (in this case, the hypotenuse) of any triangle is less than the sum of the other two sides.
Go direct.
Any ideas on the others?