Question

1. a car travels 2.3 miles east and then turns north and travels 4.1 miles. how far away is the car from the original starting point? ________ how many total miles did the car travel?______8) a ladder is leaning against a tree. the ladder hits the tree 12.2 feet above the ground and the base of the ladder is 6.3 feet from the base of the tree. how long is the ladder?______9) a square has side lengths of 6 inches. how long is the diagonal of the square?______(reduced radical)10) a rectangle has a width of 12 cm. the diagonal is 15 cm long. what is the perimeter of the rectangle?________

Explanation:

Problem 7

Step1: Find straight-line distance (Pythagoras)

Let $a=2.3$, $b=4.1$. Use $c=\sqrt{a^2+b^2}$
$\sqrt{2.3^2 + 4.1^2} = \sqrt{5.29 + 16.81} = \sqrt{22.1} \approx 4.70$

Step2: Calculate total distance traveled

Add the two distances
$2.3 + 4.1 = 6.4$

Problem 8

Step1: Find ladder length (Pythagoras)

Let $a=12.2$, $b=6.3$. Use $c=\sqrt{a^2+b^2}$
$\sqrt{12.2^2 + 6.3^2} = \sqrt{148.84 + 39.69} = \sqrt{188.53} \approx 13.73$

Problem 9

Step1: Find square diagonal (Pythagoras)

Let $s=6$. Use $d=s\sqrt{2}$
$6\sqrt{2}$

Problem 10

Step1: Find rectangle length (Pythagoras)

Let $w=12$, $d=15$. Use $l=\sqrt{d^2-w^2}$
$\sqrt{15^2 - 12^2} = \sqrt{225 - 144} = \sqrt{81} = 9$

Step2: Calculate perimeter of rectangle

Use $P=2(l+w)$
$2(9 + 12) = 2\times21 = 42$

Answer:

1. Straight-line distance: $\approx 4.70$ miles; Total miles traveled: $6.4$ miles
2. $\approx 13.73$ feet
3. $6\sqrt{2}$ inches
4. $42$ cm