Question

1. a 9 - foot pole casts a 3 - foot shadow. how tall is another pole that casts a shadow 6 feet long? a. 15 feet b. 9 feet c. 18 feet d. 12 feet 23. how do you rewrite the equation $x^{2}+y^{2}+6x - 8y=11$ in standard form? a. $(x - 3)^{2}+(y - 4)^{2}=38$ b. $(x + 3)^{2}+(y + 4)^{2}=36$ c. $(x - 3)^{2}+(y + 4)^{2}=38$ d. $(x + 3)^{2}+(y - 4)^{2}=38$ 24. what is the scale factor if a triangle with sides 4, 5, and 6 is similar to a triangle with sides 8, 10, and 12? a. 2 b. 3 c. 4 d. 1

Explanation:

22.

Step1: Set up proportion

Since the ratio of pole - height to shadow - length is the same for both poles, we can set up the proportion $\frac{h_1}{s_1}=\frac{h_2}{s_2}$, where $h_1 = 9$ feet, $s_1=3$ feet, and $s_2 = 6$ feet.

Step2: Solve for $h_2$

Cross - multiply the proportion: $h_1\times s_2=h_2\times s_1$. Substitute the known values: $9\times6 = h_2\times3$. Then $h_2=\frac{9\times6}{3}=18$ feet.

Step1: Complete the square for $x$ and $y$ terms

Given the equation $x^{2}+y^{2}+6x - 8y-11 = 0$.
For the $x$ - terms: $x^{2}+6x=(x + 3)^{2}-9$.
For the $y$ - terms: $y^{2}-8y=(y - 4)^{2}-16$.

Step2: Rewrite the equation

Substitute these into the original equation: $(x + 3)^{2}-9+(y - 4)^{2}-16-11 = 0$.
Simplify to get $(x + 3)^{2}+(y - 4)^{2}=36$.

Step1: Recall the formula for scale factor

The scale factor $k$ between two similar triangles with corresponding side - lengths $a_1,a_2,a_3$ and $b_1,b_2,b_3$ is given by $k=\frac{b_i}{a_i}$ (for any $i$). Let the first triangle have side - lengths $a_1 = 4,a_2 = 5,a_3 = 6$ and the second triangle have side - lengths $b_1 = 8,b_2 = 10,b_3 = 12$.

Step2: Calculate the scale factor

Take $\frac{b_1}{a_1}=\frac{8}{4}=2$, $\frac{b_2}{a_2}=\frac{10}{5}=2$, $\frac{b_3}{a_3}=\frac{12}{6}=2$. So the scale factor $k = 2$.

Answer:

C. 18 feet

23.