Standard form, approximation and error: Practice Questions

A recipe requires \( 250 \) grams of flour to make \( 12 \) cookies. If you want to make \( 30 \) cookies, how many grams of flour will you need?

The power \( P \) dissipated in a resistor is given by \( P = I^2 R \). The current is measured as \( I = 2.00 \pm 0.04 \) A and the resistance as \( R = 50.0 \pm 1.0 \) \( \Omega \). Calculate the approximate percentage error in the calculated value of \( P \).

A loan of $50,000 is taken out at an annual interest rate of 5.4%, compounded monthly. The loan is to be repaid over 10 years with equal monthly installments of $539.94. Calculate the remaining balance of the loan after 5 years, to the nearest dollar.

The first term of an arithmetic sequence is \( 7 \) and the common difference is \( 3 \). What is the sum of the first three terms of this sequence?

Determine the sum of the real solutions to the logarithmic equation:
\( 2 \log_2(x) - \log_2(x-1) = 3 \)

A company's revenue increased from $$ \$120,000$$ to $$ \$138,000$$ in one year. Calculate the percentage increase in revenue.

An infinite geometric sequence has a first term \( u_1 = 12 \) and a second term \( u_2 = 8 \). Calculate the sum to infinity, \( S_\infty \), of this sequence.

Solve the exponential equation $$5^{2x-3} = \frac{1}{25^{x+1}}$$ for $$x$$.

A city's population is currently $$350000$$ people. Experts predict two possible growth scenarios for the next $$10$$ years:
Scenario A: The population increases by $$8000$$ people each year.
Scenario B: The population increases by $$2\%$$ each year.
(a) Calculate the predicted population of the city after $$10$$ years under Scenario A.
(b) Calculate the predicted population of the city after $$10$$ years under Scenario B, giving your answer to the nearest whole number.
(c) Which scenario predicts a larger population after $$10$$ years, and by how many people?
(d) If a new housing development can only support a maximum population of $$450000$$ people, for which scenario (if any) will the city exceed this limit within $$10$$ years?

A logistics company manages three warehouses: X, Y, and Z. The total storage capacity across all three is \( 18000 \text{ units} \).
The capacity of warehouse Z is \( 2000 \text{ units} \) more than the combined capacity of warehouses X and Y.
Twice the capacity of warehouse X added to the capacity of warehouse Y equals the capacity of warehouse Z.

(a) Write a system of linear equations to represent the storage capacities \( x, y, \) and \( z \).
(b) Find the storage capacity of each warehouse by solving the system.