Question

question 4 of 12
undertake a full quantitative analysis of the following problem.
kelly wants to fence in a rectangular space in her yard, 9 meters by 12 meters. the salesperson at the supply store recommends that she put up posts every 1.5 meters. the posts cost $10 each. kelly will also need to buy wire mesh to string between the posts. the wire mesh is sold by the meter from large rolls and costs $7 per meter. a gate to fit in one of the spaces between posts costs $87. five staples are required to attach the wire mesh to each post. staples come in boxes of 1000, and each box costs $16.
how much will the materials cost before sales tax?
length (longer dimension) of yard: □ meters
width (shorter dimension) of yard: □ meters
distance between posts: □ meters
number of posts required: □
cost per post: □ dollars
total cost of posts: □ dollars

Explanation:

Step1: Determine yard dimensions

The yard is rectangular with sides 9m and 12m. So length (longer) is 12m, width (shorter) is 9m.
Length: $12$ meters, Width: $9$ meters

Step2: Distance between posts

Given as 1.5 meters.
Distance between posts: $1.5$ meters

Step3: Calculate perimeter of yard

Perimeter of rectangle $P = 2\times (length + width)$
$P = 2\times(12 + 9) = 2\times21 = 42$ meters

Step4: Number of posts required

Number of posts = $\frac{Perimeter}{Distance\ between\ posts}$ (since it's a closed rectangle, number of intervals = number of posts)
Number of posts = $\frac{42}{1.5} = 28$

Step5: Cost per post

Given as $10$ dollars.
Cost per post: $10$ dollars

Step6: Total cost of posts

Total cost of posts = Number of posts $\times$ Cost per post
Total cost of posts = $28\times10 = 280$ dollars

(Note: This is just the start, we need to continue for wire mesh, gate, staples. Let's complete the full analysis.)

Step7: Length of wire mesh (excluding gate)

Perimeter is 42m, gate replaces one space between posts (length 1.5m). So wire mesh length = $42 - 1.5 = 40.5$ meters

Step8: Cost of wire mesh

Cost per meter is $7$, so cost = $40.5\times7 = 283.5$ dollars

Step9: Cost of gate

Gate costs $87$ dollars.

Step10: Number of staples required

Each post needs 5 staples, 28 posts: $28\times5 = 140$ staples

Step11: Number of staple boxes

Staples come in boxes of 1000. Since 140 < 1000, we need 1 box.

Step12: Cost of staples

1 box costs $16$ dollars.

Step13: Total cost

Sum all costs: Posts ($280$) + Wire Mesh ($283.5$) + Gate ($87$) + Staples ($16$)
Total = $280 + 283.5 + 87 + 16 = 666.5$ dollars

But for the given blanks:

Length (longer dimension) of yard: $\boldsymbol{12}$ meters
Width (shorter dimension) of yard: $\boldsymbol{9}$ meters
Distance between posts: $\boldsymbol{1.5}$ meters
Number of posts required: $\boldsymbol{28}$
Cost per post: $\boldsymbol{10}$ dollars
Total cost of posts: $\boldsymbol{280}$ dollars

Answer:

(for the blanks):
Length (longer dimension) of yard: $\boldsymbol{12}$ meters
Width (shorter dimension) of yard: $\boldsymbol{9}$ meters
Distance between posts: $\boldsymbol{1.5}$ meters
Number of posts required: $\boldsymbol{28}$
Cost per post: $\boldsymbol{10}$ dollars
Total cost of posts: $\boldsymbol{280}$ dollars

(Full total cost before tax: $\boldsymbol{666.5}$ dollars)