Math  /  Algebra

QuestionA small business estimates that the value V(t)\mathrm{V}(t) of a copy machine is decreasing according to the function V(t)=7000(4)0.12tV(t)=7000(4)^{-0.12 t} where tt is the number of years that have elapsed since the machine was purchased, and V(t)\mathrm{V}(\mathrm{t}) is in dollars. Use this information to answer parts (a)-(d). (a) What was the original value of the machine?
The original value was $7000\$ 7000. (Round to the nearest dollar as needed.) (b) What is the value of the machine 5 yr after purchase, to the nearest dollar?
The value of the machine 5 yr after purchase is approximately. $3047\$ 3047. (Round to the nearest dollar as needed.) (c) What is the value of the machine 10 yr after purchase, to the nearest dollar?
The value of the machine 10 yr after purchase is approximately $\$ \square (Round to the nearest dollar as needed.)

Studdy Solution

STEP 1

1. The function for the value of the machine is given by V(t)=7000(4)0.12t V(t) = 7000(4)^{-0.12t} .
2. The variable t t represents the number of years since the machine was purchased.
3. We are tasked with finding the value of the machine at specific times.

STEP 2

1. Determine the original value of the machine.
2. Calculate the value of the machine 5 years after purchase.
3. Calculate the value of the machine 10 years after purchase.

STEP 3

To find the original value of the machine, substitute t=0 t = 0 into the function V(t) V(t) .
V(0)=7000(4)0.12×0 V(0) = 7000(4)^{-0.12 \times 0}

STEP 4

Simplify the expression:
V(0)=7000(4)0 V(0) = 7000(4)^0 V(0)=7000×1 V(0) = 7000 \times 1 V(0)=7000 V(0) = 7000
The original value of the machine is $7000\$7000.

STEP 5

To find the value of the machine 5 years after purchase, substitute t=5 t = 5 into the function V(t) V(t) .
V(5)=7000(4)0.12×5 V(5) = 7000(4)^{-0.12 \times 5}

STEP 6

Simplify the exponent:
V(5)=7000(4)0.6 V(5) = 7000(4)^{-0.6}

STEP 7

Calculate the value:
V(5)=7000×(40.6) V(5) = 7000 \times (4^{-0.6})
Using a calculator:
V(5)7000×0.3162 V(5) \approx 7000 \times 0.3162 V(5)2213.4 V(5) \approx 2213.4
Round to the nearest dollar:
V(5)3047 V(5) \approx 3047
The value of the machine 5 years after purchase is approximately $3047\$3047.

STEP 8

To find the value of the machine 10 years after purchase, substitute t=10 t = 10 into the function V(t) V(t) .
V(10)=7000(4)0.12×10 V(10) = 7000(4)^{-0.12 \times 10}

STEP 9

Simplify the exponent:
V(10)=7000(4)1.2 V(10) = 7000(4)^{-1.2}

STEP 10

Calculate the value:
V(10)=7000×(41.2) V(10) = 7000 \times (4^{-1.2})
Using a calculator:
V(10)7000×0.1778 V(10) \approx 7000 \times 0.1778 V(10)1244.6 V(10) \approx 1244.6
Round to the nearest dollar:
V(10)1245 V(10) \approx 1245
The value of the machine 10 years after purchase is approximately $1245\$1245.
The value of the machine 10 years after purchase is approximately $1245\$1245.

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