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4.
The value of a truck decreases exponentially since its purchase. The two points on the
graph shows the truck's initial
value and its value a decade afterward.
[6040,000)
a) Express the car's value, in dollars, as a function of time
d, in decades, since purchase.
(1 24,000)
b) Write an expression to represent the car's value 4 years
after purchase.
c) By what factor is the value of the car changing each year? Show your reasoning.

Sagot :

Answer:

a. v = 40 000 (3/ 5)^d

b. v = 40 000 (3/5)^(4/10)

c. 0.95

Explanation:

The exponential growth is modelled by

[tex]v=A(b)^d[/tex]

We know that points (0, 40 000) and (1, 24 000) lie on the curve. This means, the above equation must be satsifed for v = 40 000 and d = 0. Putting v = 40 000 and d = 0 into the above equation gives

[tex]40\; 000=Ab^0[/tex]

[tex]40\; 000=A[/tex]

Therefore, we have

[tex]v=40\; 000b^d[/tex]

Similarly, from the second point (1, 24 000) we put v = 24 000 and d = 1 to get

[tex]24\; 000=40\; 000b^1[/tex][tex]24\; 000=40\; 000b^{}[/tex]

dividing both sides by 40 000 gives

[tex]b=\frac{24\; 000}{40\; 000}[/tex][tex]b=\frac{3}{5}[/tex]

Hence, our equation that models the situation is

[tex]\boxed{v=40\; 000(\frac{3}{5})^d\text{.}}[/tex]

Part B.

Remember that the d in the equation we found in part A is decades. Since there are 10 years in a decade, we can write

t = 10d

or

d = t/10

Where t = number of years

Making the above substitution into our equation gives

[tex]v=40\; 000(\frac{3}{5})^{\frac{t}{10}}[/tex]

Therefore, the car's value at t = 4 is

[tex]\boxed{v=40\; 000(\frac{3}{5})^{\frac{4}{10}}}[/tex]

Part C:

The equation that gives the car's value after t years is

[tex]v=40\; 000(\frac{3}{5})^{\frac{t}{10}}[/tex]

which using the exponent property that x^ab = (x^a)^b we can rewrite as

[tex]v=40\; 000\lbrack(\frac{3}{5})^{\frac{1}{10}}\rbrack^t[/tex]

Since

[tex](\frac{3}{5})^{\frac{1}{10}}=0.95[/tex]

Therefore, our equation becomes

[tex]v=40\; 000\lbrack0.95\rbrack^t[/tex]

This tells us that the car's value is changing by a factor of 0.95 each year.

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