The Big Pumpkin, a restaurant and pie shop, is a short distance from a
major highway. The highway passes, in a straight line, through and
on a map. The Big Pumpkin is located at . What is the shortest
distance from The Big Pumpkin to the highway, to the nearest tenth of a
kilometre, if 1 unit on the map represents 1 km ? Consider a sketch.

Ask by Black Matthews. in Canada

Mar 12,2025

Question

The Big Pumpkin, a restaurant and pie shop, is a short distance from a
major highway. The highway passes, in a straight line, through and
on a map. The Big Pumpkin is located at . What is the shortest
distance from The Big Pumpkin to the highway, to the nearest tenth of a
kilometre, if 1 unit on the map represents 1 km ? Consider a sketch.

Ask by Black Matthews. in Canada

Mar 12,2025

UpStudy AI · Accepted Answer

We first determine the equation of the line representing the highway. The highway passes through $$ (-4,-2) $$ and $$ (8,4) $$. The slope $$ m $$ is given by

$$
m=\frac{4-(-2)}{8-(-4)}=\frac{6}{12}=\frac{1}{2}.
$$

Using point-slope form with the point $$ (-4,-2) $$, the equation becomes

$$
y-(-2)=\frac{1}{2}(x-(-4)) \quad \Longrightarrow \quad y+2=\frac{1}{2}(x+4).
$$

Expanding and simplifying,

$$
y+2=\frac{1}{2}x+2 \quad \Longrightarrow \quad y=\frac{1}{2}x.
$$

Next, we compute the distance from The Big Pumpkin, located at $$ (-1,3) $$, to the line $$ y=\frac{1}{2}x $$. We first rewrite the line in standard form. Multiplying both sides by 2,

$$
2y=x \quad \Longrightarrow \quad -x+2y=0.
$$

The distance $$ d $$ from a point $$ (x_0,y_0) $$ to the line $$ Ax+By+C=0 $$ is given by

$$
d=\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}.
$$

For our line, we have $$ A=-1 $$, $$ B=2 $$, and $$ C=0 $$. Substituting the coordinates of $$ (-1,3) $$:

$$
d=\frac{|-1(-1)+2(3)+0|}{\sqrt{(-1)^2+2^2}}=\frac{|1+6|}{\sqrt{1+4}}=\frac{7}{\sqrt{5}}.
$$

Calculating the numerical value,

$$
\frac{7}{\sqrt{5}}\approx\frac{7}{2.2361}\approx3.1305.
$$

Rounded to the nearest tenth, the shortest distance is approximately

$$
3.1 \text{ km}.
$$
The shortest distance from The Big Pumpkin to the highway is approximately 3.1 kilometers.

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