likewise, the front of the house is defined as the width of the house and the depth of the house is defined as the length of the house.
a..........................................................................
since the width of the lot is 91 feet and you need a minimum of 10 distance from the house to the lot line on each side of the house, then, if you let x equal the width of the house, you get:
x + 20 <= 91
subtract 20 from both sides of this equation to get:
x <= 71
this means the width of the house must be less than 71 feet.
b..........................................................................
what is described as the permissible width of the house, i have defined as permissible lengths of the house.
if the area of the house must be >= 2800 and <= 3200, and you let A represent the area of the house, then you get:
2800 <= A <= 3200
since the area of the house is equal to the length of the house * the width of the house, and you let L represent the length of the house and W represent the width of the house, then the formula becomes:
2800 <= L * W <= 3200
divided both sides of this inequality by W to get:
2800 / W <= L <= 3200 / W
the width of the house can be any value <= 71.
the formula is dependent on the width.
therefore, you have many possible values of L.
for example:
if the width of the house is 71, then the formula becomes 2800 / 71 <= L <= 3200 / 71
solve for L in this inequality and you get:
39.437 <= L <= 45.07.
if the width of the house is 20, then the formula becomes 2800 / 20 <= L <= 3200 / 20.
solve for L in this inequality and you get:
140 <= L <= 160.
so the formula is 2800 / W <= L <= 3200 / W and you need to know the specific value of W in order to solve this inequality.
this can also be solved graphically in the following manner.
let x = the width of the house
let y = the length of the house.
your constraints are:
x <= 71
x * y >= 2800
x * y <= 3200
graph the opposite of these constraints using the desmos.com calculator, and the area on the graph that is not shaded is your region of feasibility with all permissible values of x and y being between the extremes of that region.
the graph when the value of W is equal to 71 is shown below.
the graph when the value of W is equal to 20 is shown below.
the overall graph when x goes from 0 to 71 is shown below.
permissible values of L are in the area of the graph that is not shaded.
you just pick a point in the non-shaded area and the value for x and y tells you what the width of the house is and what the corresponding length of the house needs to be depending on that value for the width.
here's the overall graph.
here's the same graph with the value of x equal to 30.
the graph shows the maximum value of y and the minimum value of y.
these are at the extremes of the region of feasibility when x = 30
remember, x represents the width of the house and y represents the length of the house.
here's the same graph with the value of x equal to 30 and the value of y = 100.
you can see that, when x = 30 and y = 100, the area of the house is equal to x * y = 3000 and 3000 is between 2800 and 3200, so the area of the house is consistent with the requirements that the area of the house be greater than or equal to 2800 and less than or equal to 3200.
c..........................................................................
the cost of the house is expected to be 175,000 plus or minus 20,000.
the equation for this would be absolute value of (x - 175,000) <= 20,000
this absolute value equation is separated into 2 parts.
when (x - 175,000) is positive, the equation becomes (x - 175,000) <= 20,000.
simplify this to get x - 175,000 <= 20,000
add 175,000 to both sides of this inequality to get x <= 195,000.
if you want to stay within 20,000 from 175,000, the maximum value of the house can't be greater than 195,000.
when (x - 175,000) is negative, the equation becomes -(x - 175,000) <= 20,000.
multiply both sides of this inequality by -1 to get (x - 175,000) >= -20,000.
simplify to get x - 175,000 >= -20,000.
add 175,000 to both sides of this inequality to get x >= 155,0000.
if you want to stay with 20,000 of 175,000, the minimum value of the house can't be less than 155,000.
remember, when you multiply both sides of an inequality by a negative number, the inequality is reversed.
that's why multiplying both sides of -(x - 175,000) <= 20,000 becomes (x - 175,000) >= -20,000.