The rectangle with vertices $(-1, y), (7, y), (-1, 3)$, and $(7, 3)$ has an area of 72 square units, and $y$ is positive. What is the value of $y$?
First we write the rectangle's side lengths in terms of the coordinates provided. The length is $7-(-1)=8$ and the height is $y-3.$ It follows that $8(y-3)=72,$ and $y=\boxed{12}.$ [asy]
import graph;
size(4cm);
defaultpen(linewidth(0.7)+fontsize(10));
dotfactor=4;
xaxis(Arrows(4));
yaxis(ymin=-2,Arrows(4));
pair A=(-1,12), B=(7,12), C=(-1,3), D=(7,3);
pair[] dots = {A,B,C,D};
dot(dots);
draw(A--B--D--C--cycle);
label("$8$",(A+B)/2,N);
label("$y-3$",(B+D)/2,E);
[/asy]
12
Level 3
Geometry
Problem: The rectangle with vertices $(-1, y), (7, y), (-1, 3)$, and $(7, 3)$ has an area of 72 square units, and $y$ is positive. What is the value of $y$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the value of \( y \), we first need to determine the dimensions of the rectangle formed by the given vertices. The vertices are \( (-1, y) \), \( (7, y) \), \( (-1, 3) \), and \( (7, 3) \). 

The length of the rectangle can be calculated by finding the distance between the x-coordinates of the vertices. The x-coordinates are \( -1 \) and \( 7 \). Thus, the length \( L \) is:

\[
L = 7 - (-1) = 7 + 1 = 8
\]

Next, we need to find the height \( H \) of the rectangle. The height is determined by the difference in the y-coordinates of the vertices. The y-coordinates are \( y \) and \( 3 \). Therefore, the height \( H \) is:

\[
H = y - 3
\]

The area \( A \) of a rectangle is given by the formula:

\[
A = L \times H
\]

Substituting the known values into the area formula, we have:

\[
72 = 8 \times (y - 3)
\]

To isolate \( y - 3 \), we divide both sides by 8:

\[
y - 3 = \frac{72}{8} = 9
\]

Next, we solve for \( y \) by adding 3 to both sides:

\[
y = 9 + 3 = 12
\]

Since \( y \) is positive, the solution is valid. Thus, the value of \( y \) is \( 12 \).<|END_OF_LATENT|> The final answer is $12$.