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SIZIF [17.4K]
3 years ago
9

A 4-billion-bushel corn crop brings a price of $2.4/bushel. A commodity broker uses the following rule of thumb: If the crop is

reduced by x percent, then the price increases by 10x cents. Which crop size results in maximum revenue and what is the price per bushel? Revenue is equal to price times crop size. (Use decimal notation. Give your answers to two decimal places.) crop size__________.billion bushels price per bushel___________.$
Mathematics
1 answer:
kupik [55]3 years ago
7 0

Answer:

Step-by-step explanation:

The second crop size results in maximum revenue. The price per bushel of the second crop size is 3.54 dollars.

Step-by-step explanation: FIRST: Ensure all figures for price are in the same unit (same for quantity)

Hence, 10X cents = 0.1X dollars

SECOND:

Derive the equations for the new price and new quantity

P2 = P1 + 0.1X

Q2 = Q1 - (X% of Q1)

THIRD:

Input the values of Q1 and P1 into the equations

FOURTH:

To get X, equate P2 to Q2

P2 = Q2

From here, X = 11.4286

FIFTH:

Use this value of X to find the values of Q2 and P2 which will be 3.543 billion bushels and 3.54286 dollars respectively.

SIXTH:

The new revenue will now be 12.55 billion dollars which is obviously higher than the initial revenue of 9.6 billion dollars. Remember that R=P.Q

Revenue= Price x Quantity

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Step-by-step explanation:

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If $1,000 is invested at 4% simple interest, how much will the investment be worth after 2 years? Please explain how compound an
Novay_Z [31]
<h3>Answer:</h3>
  • simple interest: $1080.00
  • compounded annually: $1081.60
<h3>Step-by-step explanation:</h3>

<em>Simple Interest</em>

Simple interest is computed on the principal amount only. Each year, 4% of the principal is added to the balance. So, at the end of 2 years, the balance is ...

... $1000 + 0.04×$1000 + 0.04×$1000

... = $1000×(1 + 0.04×2) = $1000×1.08

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<em>Comment on the computation</em>

The added interest is the rate (per year) multiplied by the number of years. Here, that is 0.04×2×(principal amount). The formula for the simple interest earned is often seen as ...

... I = Prt . . . . . where I is the amount of interest, P is the principal amount, r is the interest rate for the time period, t is the number of time periods.

The account balance (A) with interest added is ...

... A = P + I = P + Prt

... A = P(1 +rt)

Here, the time period is years, and the rate given is an annual rate.

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<em>Compound Interest</em>

Compound interest is computed on the <em>account balance</em> at the beginning of the period, not just the <em>principal</em> amount. After the first period, the account balance includes interest earned so far. So, the interest is earning interest. That is why it is called compounded interest.

Here, the balance at the end of the first year is the principal amount plus the interest that has earned:

... $1000 + 0.04×$1000 = $1000×1.04 = $1040.00

The balance at the end of the second year when the interest is compounded is this account balance plus the interest it earns:

... $1040 + 0.04×$1040 = $1040×1.04 = $1081.60

You may notice that the intial principal, $1000, has been multiplied by the factor 1.04 twice. Using exponents, the multiplier for a period of 2 years is 1.04×1.04 = 1.04².

_____

<em>Comment on the computation</em>

The multiplier of the account balance each year is raised to a power that is the number of years. Here, the account balance at the end of 2 years is (1+0.04)² times the principal amount. A formula that is seen for this is ...

... A = P(1 +r)^t . . . . . where A is the final account balance, P is the principal amount, r is the interest rate for the time period, and t is the number of time periods.

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