SOLUTION: . A construction company purchases a bulldozer for $160.000. Each year the value of the bulldozer depreciates by 20% of its value in the preceding year. Let V, be the value of the

Algebra.Com
Question 1209756: . A construction company purchases a bulldozer for $160.000. Each year the value of the bulldozer depreciates by 20% of its value in the preceding year. Let V, be the value of the bulldozer in the nth year. (Let n = 1 be the year the bulldozer is purchased)
a) Find the formula for V.
12 Marks)
b) In what year will the value of the bulldozer be less than $100.000
(4 Marks)

Found 2 solutions by CPhill, Edwin McCravy:
Answer by CPhill(1959)   (Show Source): You can put this solution on YOUR website!
Here's how to solve this depreciation problem:
**a) Formula for V:**
* **Initial Value:** The bulldozer starts at $160,000.
* **Depreciation Rate:** The bulldozer depreciates by 20% (0.20) each year.
* **Value After 1 Year:** $160,000 * (1 - 0.20) = $160,000 * 0.80 = $128,000
* **Value After 2 Years:** $128,000 * 0.80 = $160,000 * 0.80 * 0.80 = $160,000 * (0.80)²
* **General Formula:** V(n) = $160,000 * (0.80)^(n-1)
**b) Year When Value is Less Than $100,000:**
We want to find the smallest integer *n* such that V(n) < $100,000.
1. **Set up the inequality:**
$160,000 * (0.80)^(n-1) < $100,000
2. **Divide by 160,000:**
(0.80)^(n-1) < 100,000 / 160,000
(0.80)^(n-1) < 0.625
3. **Take the logarithm (base 10 or natural log) of both sides:**
log(0.80)^(n-1) < log(0.625)
(n-1) * log(0.80) < log(0.625)
4. **Divide by log(0.80). Note that log(0.80) is negative, so reverse the inequality sign:**
n - 1 > log(0.625) / log(0.80)
n - 1 > -0.2076 / -0.0969
n - 1 > 2.1424
5. **Add 1 to both sides:**
n > 3.1424
6. **Find the smallest integer:**
Since *n* must be an integer, the smallest value of *n* that satisfies this inequality is *n* = 4.
**Answer:**
a) The formula for V is V(n) = $160,000 * (0.80)^(n-1)
b) The value of the bulldozer will be less than $100,000 in the 4th year.

Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
Looks like AI can solve this type problem. Give AI a few years and it'll be
able to solve any mathematics problem. In the beginning, anybody could beat
AI in chess. But the last time a human could beat AI in chess was in 2008.
It hasn't quite gotten there in mathematics. But give it time and it will!
[In the US, we use a DOT "." for a decimal point, and COMMAS "," to separate
digits in groups of three. It is just the opposite in your country. But nobody
separates digits in groups of three when calculating.]

Since the bulldozer drops in value by 20% each year means that each year, its
value is only 80% of what it was the year before.  So, we are talking about the
geometric sequence with first term a1=160,000, and common ratio
r=0.80, and its nth term is an = vn. 

(a)
160,000, (0.80)(160000), (0.80)2(160000), (0.80)3(140000),...

 



(b)
Since it will only take a few years, it's easier to do it this way
The 1st year the value is $160,000.
The 2nd year the value is $160,000(0.80) = $128,000.
The 3rd year the value is $128,000(0.08) = $102,400.
The 4th year the value is $102,000(0.80) = $81,920.
So the 4th year is the first year its value will be less than $100,000 all year.

But your teacher might expect you to use the formula. So you'd do it
this way to please your teacher:












So the year after 3.10628372 years its value will be less than $100,000.
That means that the 4th year is the first year its value will be less 
than $100,000 all year.

Edwin

RELATED QUESTIONS

A construction company purchases a bull- dozer for $160,000. Each year the value of the... (answered by Boreal)
Depreciation is the decrease or loss in value of an item due to age, wear, or market... (answered by addingup)
Question A) A company needs to clear 400ha of land. The company has newly acquired a D7 H (answered by akomfestus@gmail.com)
A bulldozer was purchased by a construction company for $224,000 and has a depreciated... (answered by Boreal)
I am having trouble with this word problem. Ace Construction Company rents a... (answered by checkley79)
A musical instrument depreciates by 20% of its value each year. What is the value, after (answered by Alan3354)
It is estimated that a bulldozer can prepare a certain site in 15.0 days and that a... (answered by mananth,josgarithmetic)
This is a Value Depreciation Model A company purchases a complete industrial... (answered by Theo)
Hi, hope you can help me withs this problem A new car depreciates by 20% in its first... (answered by stanbon)