I have a 3 part question that I have the answers for but I am still having issues understanding it. Looking for help with the explanation / steps on how you would get the answer.
Question
1) The width of a rectangular field is 3h yards and it's length is 3 yards longer than the width. The field has a fence around it's perimeter with a gate 4 yards wide.
A) Write an expression for the perimeter of the rectangular field in terms of h, excluding the width of the gate.
Answer
(12 h + 2 ) yards
B) It costs $28 per yard to fence the field, excluding the gate. Write an expression that represents the cost of fencing the field.
Answer
The cost is 28 (12h + 2 ) dollars
C) If h = 5, find the cost of the fencing, excluding the gate.
Answer
The cost is $1,736
width ---- 3h
length ---- 3h+3
perimeter = 2(3h) + 2(3h+3) - 4
= 6h + 6h + 6 - 4
= 12h +2
you are correct
b) correct 28(12h+2)
=
c) cost = 28(12h+2)
= 28(60+2) = 1736
you are correct.
good job
Thank you
To understand the steps and explanations for solving the three questions, let's break it down:
Question 1: The width of a rectangular field is 3h yards, and its length is 3 yards longer than the width. The field has a fence around its perimeter with a gate 4 yards wide.
A) Write an expression for the perimeter of the rectangular field in terms of h, excluding the width of the gate.
To find the perimeter of the rectangular field, we need to add up all the sides of the rectangle, excluding the gate. The gate width is given as 4 yards, so we need to subtract that from the total.
Perimeter of the rectangular field = 2(Length + Width) - Gate Width
= 2(3h + 3) - 4
= 6h + 6 - 4
= 6h + 2
Therefore, the expression for the perimeter of the rectangular field in terms of h, excluding the width of the gate, is (6h + 2) yards.
B) It costs $28 per yard to fence the field, excluding the gate. Write an expression that represents the cost of fencing the field.
To find the cost of fencing the field, we need to multiply the perimeter of the rectangular field (excluding the gate) by the cost per yard.
Cost of fencing the field = Cost per yard * Perimeter of the rectangular field (excluding the gate)
= 28 * (6h + 2)
= 28(6h) + 28(2)
= 168h + 56
Therefore, the expression that represents the cost of fencing the field, excluding the gate, is (168h + 56) dollars.
C) If h = 5, find the cost of the fencing, excluding the gate.
To find the cost of fencing, excluding the gate, when h is equal to 5, we substitute h = 5 in the expression we derived in part B.
Cost of fencing = 168h + 56 (when h = 5)
= 168(5) + 56
= 840 + 56
= 896
Therefore, the cost of the fencing, excluding the gate, when h is equal to 5, is $896.
To explain how to get the answers to each part of the question, let's break it down step by step.
1) Part A: Writing an expression for the perimeter of the rectangular field in terms of h, excluding the width of the gate.
In this part, we need to find the perimeter of the field without considering the width of the gate.
The width of the rectangular field is given as 3h yards, which means the length of the field is 3 yards longer than the width. So, the length would be 3h + 3 yards.
The formula for the perimeter of a rectangle is P = 2(length + width). In this case, excluding the gate, we have P = 2(3h + 3 + 3h).
Simplifying the expression:
P = 2(6h + 3)
P = 12h + 6
However, since we are excluding the width of the gate, we subtract 4 yards from the expression.
P = 12h + 6 - 4
P = 12h + 2 yards.
Therefore, the expression for the perimeter of the rectangular field in terms of h, excluding the width of the gate, is (12h + 2) yards.
2) Part B: Writing an expression that represents the cost of fencing the field, excluding the gate.
The cost to fence the field is given as $28 per yard, excluding the gate. Therefore, we need to multiply the cost per yard by the total perimeter of the field (excluding the gate).
Using the expression we found in part A, the cost of fencing the field would be:
Cost = 28(12h + 2) dollars.
Therefore, the expression that represents the cost of fencing the field, excluding the gate, is 28(12h + 2) dollars.
3) Part C: Finding the cost of the fencing, excluding the gate, when h = 5.
To find the cost of the fencing when h = 5, we substitute h = 5 into the expression we found in part B.
Cost = 28(12h + 2)
Cost = 28(12*5 + 2)
Cost = 28(60 + 2)
Cost = 28(62)
Cost = $1,736.
So, when h = 5, the cost of the fencing, excluding the gate, is $1,736.