IB Maths AI SL AI 1.1 practice question

Question 1

Paper 1Hard16 marks

A workshop makes a circular cover. Its radius is measured as $6.4\text{ cm}$ correct to the nearest $0.1\text{ cm}$ The workshop estimates the area using this displayed radius and rounds the estimate to three significant figures for its label.

For all parts, use unrounded calculator values in subsequent calculations unless a specified rounding is requested. Percentage error is defined as

\frac{|\text{estimate}-\text{true value}|}{\text{true value}}\times 100\%.

(a)

Calculate the estimated area of the cover, giving your answer to three significant figures.

[2]

(b)

Write down the lower and upper bounds for the radius.

[2]

(c)

Let $A\text{ cm}^2$ be the true area of the cover. Show that a guaranteed interval for $A$ is

126.67\leq A<130.70.

[3]

(d)

Calculate the maximum percentage error in the area estimate.

[3]

(e)

The workshop prints $129\text{ cm}^2$ on the cover’s label. Determine whether this label is guaranteed to have a percentage error of less than $2\%$ Justify your answer.

[2]

(f)

A new gauge displays the radius as $6.42\text{ cm}$ correct to the nearest $q\text{ cm}$ Determine the greatest possible value of $q$ for which the estimated area is guaranteed to have a percentage error of at most $1\%$ State whether rounding to the nearest $0.1\text{ cm}$ and to the nearest $0.01\text{ cm}$ meet this requirement.

[4]