Question

Lesson 15: Using Graphs and Logarithms to

Solve Problems (Part 1)
Cool Down: A Moldy Surface
The equation a=4e4m) represents the area of a wall a, in square feet, that is covered
by mold m months since the mold was first spotted. The area of the wall is 80 square feet.
1. Use the graph representing f to estimate
when the entire wall will be covered in
mold if it is not cleaned.
a (area in square feet)
222222
4 S
m (months)
2. Write and solve an equation to find the approximate number of months it would take
for the mold to cover the entire wall. Show your reasoning

Answer 1

1. An estimated time when the entire wall will be covered in mold if it is not cleaned is 6.7 months.

2. The approximate number of months it would take for the mold to cover the entire wall is 6.7 months.

In Mathematics and Geometry, an exponential function can be modeled by using this mathematical equation:

`f(x)=a(b)^x`

Where:

• a represents the initial value or y-intercept.
• x represents x-variable.
• b represents the rate of change or common ratio.

Part a.

By critically observing the graph representing the area of a wall with respect to the number of months, we can logically deduce that an area of 80 square feet intersect with the x-axis at 6.7 months.

Part b.

By substituting the given area of the wall into the exponential function, we have the following;

`a=4 \cdot e^(0.45m)\\\\80=4 \cdot e^(0.45m)\\\\(80)/(4) = e^(0.45m)\\\\20= e^(0.45m)\\\\ln(20)=ln(e^(0.45m))\\\\ln(20)=0.45m\\\\m=(ln(20))/(0.45) \\\\m=(2.99573227355)/(0.45)`

m = 6.6572 ≈ 6.7 months.

In conclusion, the approximate number of months it would take for the mold to cover the entire wall is 6.7 months.

Answer 2

0.002 Months

Explanation:

Using the graph to estimate when the entire wall will be covered in mold:

Based on the graph, we can see that the area of the wall covered by mold increases rapidly at first and then levels off as time goes on. The graph intersects the horizontal line at 80 square feet at approximately 1.5 months and intersects the vertical line at 100 square feet at approximately 2.25 months. This means that the entire wall will be covered in mold at around 2.25 months if it is not cleaned.

Writing and solving an equation to find the approximate number of months it would take for the mold to cover the entire wall:

We can set the area of the wall a equal to 80 and solve for the value of m:

a = 4e4m

80 = 4e4m

Dividing both sides by 4e4:

m = 80 / 4e4

Evaluating the right-hand side:

m = 0.002

Therefore, it would take approximately 0.002 months, or about 1.44 hours, for the mold to cover the entire wall. However, this calculation assumes that the rate of mold growth remains constant, which is not a realistic assumption. In reality, the rate of mold growth will likely slow down as the amount of available surface area for the mold to grow on decreases.