IB myp 4-5 MATHEMATICS – Practice Questions- All Topics
Topic :Geometry–Volume and capacity
Topic :Geometry- Weightage : 21 %
All Questions for Topic : Volume and capacity (additional shapes),Enlargement around a given point,Enlargement by a rational factor,Gradients of perpendicular lines,Identical representation of transformations
Question
Question (a)
Show that $r=2.80 \mathrm{~cm}$, correct to three significant figures.
▶️Answer/Explanation
Ans:
To solve the equation $\cos 15^\circ = \frac{2.7}{r}$ for $r$, we can rearrange the equation to isolate $r$ on one side:
$\cos 15^\circ = \frac{2.7}{r}$
Multiplying both sides by $r$:
$r \cos 15^\circ = 2.7$
Dividing both sides by $\cos 15^\circ$:
$r = \frac{2.7}{\cos 15^\circ}$
Now, let’s calculate the value of $r$:
$r = \frac{2.7}{\cos 15^\circ} \approx \frac{2.7}{0.965925}$ (using the cosine value of $15^\circ$)
$r \approx \frac{2.7}{0.965925} \approx 2.798$ (rounding to three significant figures)
Therefore, we find that $r \approx 2.8 \, \mathrm{cm}$.
Question (b)
The whole sphere of ice melted in the cone, as shown in the diagram below.
Find the value of $h$.
▶️Answer/Explanation
Ans:
Given:
Radius of the melted sphere, $r = 2.80 \, \mathrm{cm}$
Radius of the cone, $R = 2.86 \, \mathrm{cm}$
We can still use the concept of equal volumes to find the height $h_{\text{cone}}$.
The volume of the cone is given by:
$V_{\text{cone}} = \frac{1}{3} \pi R^2 h_{\text{cone}}$
The volume of the sphere is given by:
$V_{\text{sphere}} = \frac{4}{3} \pi r^3$
Setting the volumes equal to each other:
$\frac{1}{3} \pi R^2 h_{\text{cone}} = \frac{4}{3} \pi r^3$
Canceling out $\pi$ and simplifying, we have:
$R^2 h_{\text{cone}} = 4r^3$
Substituting the values $r = 2.80 \, \mathrm{cm}$ and $R = 2.86 \, \mathrm{cm}$, we get:
$(2.86)^2 h_{\text{cone}} = 4(2.80)^3$
$8.1796 h_{\text{cone}} = 87.5456$
Dividing both sides by $8.1796$, we find:
$h_{\text{cone}} \approx \frac{87.5456}{8.1796} \approx 10.69$
Therefore, the height of the melted ice in the cone is approximately $10.7 \, \mathrm{cm}$.