Mark’s brother is five years older than Mark. Their combined ages add up to 23. How old is Mark? Write an equation to help you s
olve.
1 answer:
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Part I: Scale Drawing
<span>Decide on a scale factor to represent the distance covered by the patio. Then, use the space below to design Mrs. Johnson’s patio to be a square that is at least 8 feet on each side.</span>
a) Scale Factor: 1 in/ 2 ft
Use a straightedge and a ruler to draw to scale a design for Mrs. Johnson’s patiob) see the picture attached
Mrs. Johnson’s patio to be a square that is 10 ft x 10 ft
<span><span>c) What are the dimensions of Mrs. Johnson’s patio? </span>
</span>the dimensions of Mrs. Johnson’s patio are 10 ft x 10 ft
<span><span>d) Calculate the area of Mrs. Johnson’s patio. Show all work.
</span> </span>
area of the square=b²
where b is the length side of the square
b=10 ft
so
Area=10²-----> area =100 ft²
<span><span>e) How many pavers will be needed? Show all work. </span>
</span>
we know that
1 paver is 16 in x 16 in dimensions
convert to ft
1 ft----------->12 in
x ft-----------> 16 in
x=16/12-----> x=4/3 ft
so
1 paver is (4/3) ft x (4/3) ft dimensions
area of one paver=(4/3)²----> 16/9 ft²
if one paver has an area of----------------> 16/9 ft²
x pavers-----------------------> 100 ft²
x=100/(16/9)------> x=100*9/16-----> x=56.25 pavers
if one box --------------> 12 pavers
x box---------> 56.25 pavers
x=56.25/12-----> x=4.68 box-------> x=5 boxes
5 boxes of pavers will be needed
<span><span>f) What will it cost to build the patio? Show all work.
</span> </span>
the cost of one box is--------> $99.99
5 boxes-----------> x
x=5*$99.99------>x=$499.95
the cost to build the patio is $499.95
Part II: Bigger Design
<span>There is a saying that bigger is better, so why not double the dimensions of Mrs. Johnson’s patio to make the side measurement twice as big? Mrs. Johnson and I think that it would better meet her needs. After seeing the original estimation, she thinks that she could afford to double the size. I explained that making the patio twice as big would mean twice the cost. Mrs. Johnson says, “Let’s do it!”</span>
<span>a) What would be the new dimensions of Mrs. Johnson’s patio?
</span>the new dimensions of Mrs. Johnson’s patio are 20 ft x 20 ft
<span>b) Calculate the new area of Mrs. Johnson’s patio. Show all work.
</span>area of the square=b²
where b is the length side of the square
b=20 ft
so
Area=20²-----> area =400 ft²<span>
</span>
<span><span>c) How many pavers will be needed for the new design? Show all work.
</span> </span>
1 paver is (4/3) ft x (4/3) ft dimensions
area of one paver=(4/3)²----> 16/9 ft²
if one paver has an area of----------------> 16/9 ft²
x pavers-----------------------> 400 ft²
x=400/(16/9)------> x=400*9/16-----> x=225 pavers
if one box --------------> 12 pavers
x box---------> 225 pavers
x=225/12-----> x=18.75 box-------> x=19 boxes
19 boxes of pavers will be needed
<span>d) What will it cost to build the bigger patio? Show all work.
</span>the cost of one box is--------> $99.99
19 boxes-----------> x
x=19*$99.99------>x=$1899.81
the cost to build the bigger patio is $1899.81
<span><span>e) Is Khianna right? Will doubling the size of the patio, double the cost?</span>
</span>
<span>Khianna is wrong to double the dimensions the cost quadruples</span>
A circle with center and radius
has equation
If we substitute in this equation, we have
This equation has two solutions (i.e. the line intersects the circle in two points) if and only if the determinant is greater than zero:
The expression simplifies to
The solutions to the associated equation are
So, the parabola is negative between the two solutions: