1. it was 19 kilometers from the campgrounds to the lake. joel and mario rode their bicycles from the…

1. it was 19 kilometers from the campgrounds to the lake. joel and mario rode their bicycles from the campgrounds to the lake and back again. if the round - trip ride took them 2 hours, what was their average speed in kilometers per hour? (53) 2. the vertices of two angles of a triangle are (3, 1) and (0, - 4). the y - axis is a line of symmetry of the triangle. a. what are the coordinates of the third vertex of the triangle? b. what is the area of the triangle? (37, 58) * 3. verify using a tape measure, gretchen found that the circumference of the great oak was 600 cm. rounding π to 3, she estimated that the tree’s diameter was 200 cm. was her estimate for the diameter a little too large or a little too small? why? (66)

1. it was 19 kilometers from the campgrounds to the lake. joel and mario rode their bicycles from the campgrounds to the lake and back again. if the round - trip ride took them 2 hours, what was their average speed in kilometers per hour? (53) 2. the vertices of two angles of a triangle are (3, 1) and (0, - 4). the y - axis is a line of symmetry of the triangle. a. what are the coordinates of the third vertex of the triangle? b. what is the area of the triangle? (37, 58) * 3. verify using a tape measure, gretchen found that the circumference of the great oak was 600 cm. rounding π to 3, she estimated that the tree’s diameter was 200 cm. was her estimate for the diameter a little too large or a little too small? why? (66)

Answer

Explanation:

Step1: Calculate average - speed for question 1

Average speed formula is $v=\frac{d}{t}$, where $d$ is distance and $t$ is time. The round - trip distance $d = 19$ km and time $t=2$ h. So, $v=\frac{19}{2}=9.5$ km/h.

Step2: Find the third vertex for question 2a

If the $y$ - axis is a line of symmetry, for a point $(x,y)$ its symmetric point about the $y$ - axis is $(-x,y)$. Given points $(3,1)$ and $(0, - 4)$, the symmetric point of $(3,1)$ about the $y$ - axis is $(-3,1)$.

Step3: Calculate the area of the triangle for question 2b

We can use the formula for the area of a triangle given the coordinates $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$: $A=\frac{1}{2}\left|x_1(y_2 - y_3)+x_2(y_3 - y_1)+x_3(y_1 - y_2)\right|$. Let $(x_1,y_1)=(3,1)$, $(x_2,y_2)=(0, - 4)$, $(x_3,y_3)=(-3,1)$. Then $A=\frac{1}{2}\left|3(-4 - 1)+0(1 - 1)+(-3)(1+4)\right|=\frac{1}{2}\left|3\times(-5)+0+(-3)\times5\right|=\frac{1}{2}\left|-15 - 15\right|=\frac{1}{2}\times30 = 15$ square units.

Step4: Check the diameter estimate for question 3

The formula for the circumference of a circle is $C=\pi d$. Given $C = 600$ cm and $\pi\approx3$, then $d=\frac{C}{\pi}=\frac{600}{3}=200$ cm. Since the actual value of $\pi\approx3.14>3$, when we use $\pi = 3$ in the formula $d=\frac{C}{\pi}$, the estimated diameter is a little too large.

Answer:

  1. 9.5 km/h
  2. a. (-3,1) b. 15 square units
  3. Her estimate for the diameter was a little too large because the actual value of $\pi\approx3.14$ and she used $\pi = 3$ in the formula $d=\frac{C}{\pi}$.