Question

Write an expression for the value of a commodity after 20 years if its value decreases at a monthly rate of 2%.​

Answer 1

ᴀ) ᴛʜᴇ ᴀɴsᴡᴇʀ ɪs ғ(ɴ) = 1500 * 0.8ⁿ

ʟᴇᴛ ᴛʜᴇ ғᴜɴᴄᴛɪᴏɴ ʙᴇ ғ(ɴ) ᴡʜᴇʀᴇ ɴ ɪs ɴᴜᴍʙᴇʀ ᴏғ ʏᴇᴀʀs

ᴀғᴛᴇʀ 20% ɪs ʟᴏsᴛ, ᴛʜᴇ ᴠᴀʟᴜᴇ ᴛʜᴀᴛ ʀᴇᴍᴀɪɴs ɪs 80% ᴏғ 1500:

1500 : 100%

x : 80%

1500 : 100% = x : 80%

x = 1500 * 80% : 100%

x = 1500 * 80/100

x = 1500 * 0.8

ғᴏʀ ᴇᴀᴄʜ ʏᴇᴀʀ ᴡᴇ ᴍᴜsᴛ ᴀᴅᴅ ᴇxᴘᴏɴᴇɴᴛ, sᴏ ᴛʜᴇ ғᴜɴᴄᴛɪᴏɴ ɪs:

ғ(ɴ) = 1500 * 0.8ⁿ

ʙ) ᴛʜᴇ ᴀɴsᴡᴇʀ ɪs $768

ᴡᴇ ʜᴀᴠᴇ ғᴜɴᴄᴛɪᴏɴ: ғ(ɴ) = 1500 * 0.8ⁿ

ᴀɴᴅ ᴡᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ ᴛʜᴇ ɴᴜᴍʙᴇʀ ᴏғ ʏᴇᴀʀs ɪs 3: ɴ = 3

ғ(3) = 1500 * 0.8³ = 1500 * 0.512 = 768

ᴄ) ᴛʜᴇ ᴀɴsᴡᴇʀ ɪs 5 ʏᴇᴀʀs

ᴡᴇ ʜᴀᴠᴇ ғᴜɴᴄᴛɪᴏɴ: ғ(ɴ) = 1500 * 0.8ⁿ

ᴀɴᴅ ᴡᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ ᴛʜᴇ ᴠᴀʟᴜᴇ ɴᴇᴇᴅs ᴛᴏ ʙᴇ ʟᴇss ᴛʜᴇɴ $500: ғ(ɴ) = <500

1500 * 0.8ⁿ < 500

0.8ⁿ < 500 / 1500

0.8ⁿ < 0.33

ʟᴏɢ(0.8ⁿ) < ʟᴏɢ(0.33)

ɴ * ʟᴏɢ(0.8) < ʟᴏɢ(0.33)

ɴ * -0.0969 < -0.4815

ɴ < -0.4815 / -0.0969

ɴ < 5