Question

1. hanna is building a raised vegetable bed in her garden, which is on a square grid where each unit is 25 cm. the vertices of the vegetable bed are at a(-1,1), b(4,2), c(3, - 2), and d(-2,-3). a) if the wooden slats she is going to use to build the perimeter cost $4.85 per meter, how much will it cost her to build the bed? t:5 b) hanna has put a tomato stake at o(0,0). what is the shortest distance from the stake to dc? t:3

Explanation:

Step1: Recall distance - formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Identify points for distance from $O(0,0)$ to $D(-2,-3)$

Here $x_1 = 0,y_1 = 0,x_2=-2,y_2 = - 3$.

Step3: Substitute values into formula

$d=\sqrt{(-2 - 0)^2+(-3 - 0)^2}=\sqrt{(-2)^2+(-3)^2}=\sqrt{4 + 9}=\sqrt{13}\approx3.61$ cm.

Step4: Recall perimeter - formula for a polygon

For a polygon with vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$, the perimeter $P=d_{12}+d_{23}+d_{34}+d_{41}$, where $d_{ij}$ is the distance between points $(x_i,y_i)$ and $(x_j,y_j)$.

Step5: Calculate distances between consecutive vertices

• Distance between $A(-1,1)$ and $B(4,2)$: $d_{AB}=\sqrt{(4 + 1)^2+(2 - 1)^2}=\sqrt{25 + 1}=\sqrt{26}\approx5.10$ cm.
• Distance between $B(4,2)$ and $C(3,-2)$: $d_{BC}=\sqrt{(3 - 4)^2+(-2 - 2)^2}=\sqrt{1+16}=\sqrt{17}\approx4.12$ cm.
• Distance between $C(3,-2)$ and $D(-2,-3)$: $d_{CD}=\sqrt{(-2 - 3)^2+(-3 + 2)^2}=\sqrt{25 + 1}=\sqrt{26}\approx5.10$ cm.
• Distance between $D(-2,-3)$ and $A(-1,1)$: $d_{DA}=\sqrt{(-1+2)^2+(1 + 3)^2}=\sqrt{1 + 16}=\sqrt{17}\approx4.12$ cm.

Step6: Calculate perimeter

$P=\sqrt{26}+\sqrt{17}+\sqrt{26}+\sqrt{17}=2\sqrt{26}+2\sqrt{17}\approx2\times5.10+2\times4.12=10.20 + 8.24=18.44$ cm.

Step7: Calculate cost

The cost per centimeter is $\$4.85$. So the total cost $C = 4.85\times18.44\approx\$89.43$.

Answer:

The shortest distance from the stake at $O$ to $D$ is $\sqrt{13}\approx3.61$ cm. The cost to build the bed is approximately $\$89.43$.