Exponetial Functions 連答案 (40分)

tfzoofss

2011-11-13 4:45 am

(21) 8^(x-1) + 2^(3x+1) = 34

(23) 4^(2x) = (2^(x-1))/(8)

(45) 3^(x+3) - 2(3^(x+2)) = 30 - 3^x

(50) Simultaneous equation
3^(x+1) - 2^(y+1) = 1
4(3^x) + 3(2^y) = 24

(28) The amount (A mg) of a certain drug in patient's body is given by A = 25(0.85^t), where t is time (in hours) after the drug is taken

(28a) Find the initial amount of drug in the patient's body

(28b) What % of drug is still in the patient's body after 4 hours (3 sig. fig.)

(29) Mr. Chan deposits $25000 in a bank at 4% p.a. compounded half-yearly. Suppse the amount received after n years is $A

(29a) Express A in terms of n

(29b) Find the amount Mr Chan will receive 3 years later (correct to nearest $1)

(54) The volume of water V (in cm^2) in a water tank t minutes after turning on the tap of the tank is given by V = A(0.94^t), where A is a constant. It is given that the volume of water in the tank is 4418 cm^3 two minutes after turning on the tap

(54a) Find the value of A

(54b) Find the volume of water running out of the tank in the 5th minute (3 sig. fig.)

Answers:
(21) 4/3
(23) -(4/3)
(45) 1
(50) x = 1, y = 2
(28a) 25 mg
(28b) 52.5%
(29a) A = 25000 (1.02)^(2n)
(29b) $28154
(54a) 5000
(54b) 234 cm^3

更新1:

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回答 (1)

wanszeto

2011-11-14 8:51 am

✔ 最佳答案

(21)
8^(x - 1) + 2^(3x + 1) = 34
(2^3)^(x - 1) + 2^(3x + 1) = 34
2^(3x - 3) + 2^(3x + 1) = 34
2^(3x - 3) + 2^[4 +(3x - 3)] = 34
2^(3x - 3) + (2^4)[2^(3x - 3)] = 34
2^(3x - 3) + 16[2^(3x - 3)] = 34
(1 + 16)[2^(3x - 3)] = 34
17[2^(3x - 3)] = 34
2^(3x - 3) = 2
2^(3x - 3) = 2^1
3x - 3 = 1
3x = 4
x = 4/3

(23)
4^(2x) = [2^(x - 1)]/8
(2^2)^(2x) = [2^(x - 1)]/(2^3)
2^(4x) = 2^[(x - 1) - 3]
2^(4x) = 2^(x - 4)
4x = x - 4
3x = -4
x = -4/3

(45)
3^(x + 3) - 2[3^(x + 2)] = 30 - 3^x
3^(x + 3) - 2[3^(x + 2)] + 3^x = 30
(3^3)(3^x) - 2(3^2)(3^x) + 3^x = 30
27(3^x) - 18(3^x) + 3^x = 30
(27 - 18 + 1)(3^x) = 30
10(3^x) = 30
3^x = 3
3^x = 3^1
x = 1

(50)
3^(x + 1) - 2^(y + 1) = 1 ... [1]
4(3^x) + 3(2^y) = 24 ... [2]

From [1]:
3(3^x) - 2(2^y) = 1
9(3^x) - 6(2^y) = 3 ... [3]

[2]*2:
8(3^x) + 6(2^y)= 48 ... [4]

[3] + [4]:
17(3^x) = 51
3^x = 3
x =1

Put x = 1 into [1]:
3^(1 + 1) - 2^(y + 1) = 1
9 - 2^(y + 1) = 1
2^(y + 1) = 8
2^(y + 1) = 2^3
y + 1 = 3
y =2

28(a)
A = 25(0.85^t)

When t = 0:
A = 25(0.85^0)
A = 25
Hence, initial amount of drug = 25 mg

(28b)
When t = 4:
A = 25(0.85^4)
A = 13.05

Percentage
= (13.05/25) * 100%
= 52.2%
(The given answer 52.5% is incorrect.)

(29a)
Amount = $A
Principal = $25000
Interest rate per period = 4%/2 = 2%
Number of periods = 2n

A = 25000 * (1 + 2%)^(2n)
Hence, A = 25000 * 1.02^(2n)

(29b)
When n = 3:
A = 25000 * 1.02^(2*3)
A = 28154
Amount received = $28154 (to nearest $)

(54a)
V = A(0.94^5)

When t = 2:
4418 = A(0.94^2)
A = 4418/(0.94^2)
A = 5000

(54b)
Water running out in the 5th minute
= [5000(0.94^4) - 5000(0.94)^5] cm^3
= 234.224688 cm^3
= 234cm^3 (3 sig. fig.)

參考: wanszeto

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