Question

1. a triangle has an area of \\((3\sqrt{288} - 2\sqrt{12})\\) square metres with a base of \\(3\sqrt{2}\\) metres

express the height of the triangle
a) as an exact value in simplest form
b) as a decimal to the nearest 0.01 m.

Explanation:

Part (a)

Step 1: Recall the area formula for a triangle

The area \( A \) of a triangle is given by \( A=\frac{1}{2} \times \text{base} \times \text{height} \), so we can solve for height \( h \) as \( h = \frac{2A}{\text{base}} \).

Step 2: Simplify the area expression

First, simplify the area \( A = 3\sqrt{288}-2\sqrt{12} \).
Simplify \( \sqrt{288} \): \( \sqrt{288}=\sqrt{144\times2} = 12\sqrt{2} \), so \( 3\sqrt{288}=3\times12\sqrt{2}=36\sqrt{2} \).
Simplify \( \sqrt{12} \): \( \sqrt{12}=\sqrt{4\times3}=2\sqrt{3} \), so \( 2\sqrt{12}=2\times2\sqrt{3} = 4\sqrt{3} \). Wait, no, wait, the problem is about a triangle with base in terms of \( \sqrt{2} \), maybe I made a mistake. Wait, let's re - check. Wait, \( \sqrt{288}=\sqrt{144\times2}=12\sqrt{2} \), so \( 3\sqrt{288}=3\times12\sqrt{2} = 36\sqrt{2} \). \( \sqrt{12}=\sqrt{4\times3}=2\sqrt{3} \), but the base is \( 3\sqrt{2} \). Wait, maybe there is a mistake in my initial simplification. Wait, no, let's do it again.

Wait, the area \( A=3\sqrt{288}-2\sqrt{12} \). Let's simplify \( \sqrt{288} \): \( 288 = 144\times2 \), so \( \sqrt{288}=12\sqrt{2} \), so \( 3\sqrt{288}=3\times12\sqrt{2}=36\sqrt{2} \). \( \sqrt{12}=\sqrt{4\times3}=2\sqrt{3} \), so \( 2\sqrt{12}=4\sqrt{3} \). But the base is \( 3\sqrt{2} \). Wait, maybe the problem has a typo? Wait, no, maybe I misread. Wait, the area is \( (3\sqrt{288}-2\sqrt{12}) \) square meters and base is \( 3\sqrt{2} \) meters.

Now, substitute into the height formula \( h=\frac{2A}{\text{base}} \).

First, calculate \( 2A \): \( 2(3\sqrt{288}-2\sqrt{12})=6\sqrt{288}-4\sqrt{12} \)

Simplify \( \sqrt{288} = 12\sqrt{2} \), so \( 6\sqrt{288}=6\times12\sqrt{2}=72\sqrt{2} \)

Simplify \( \sqrt{12}=2\sqrt{3} \), so \( 4\sqrt{12}=8\sqrt{3} \). Wait, but the base is \( 3\sqrt{2} \), so if we have \( 72\sqrt{2}-8\sqrt{3} \) divided by \( 3\sqrt{2} \), that seems complicated. Wait, maybe I made a mistake in the problem interpretation. Wait, maybe the area is \( 3\sqrt{288}-2\sqrt{12} \), but let's check the numbers again. Wait, \( \sqrt{288}=12\sqrt{2} \), \( \sqrt{12} = 2\sqrt{3} \), but maybe the second term is \( 2\sqrt{18} \) instead of \( 2\sqrt{12} \)? No, the problem says \( 2\sqrt{12} \). Wait, maybe I miscalculated. Wait, let's proceed.

Wait, \( h=\frac{2(3\sqrt{288}-2\sqrt{12})}{3\sqrt{2}}=\frac{6\sqrt{288}-4\sqrt{12}}{3\sqrt{2}} \)

Simplify \( \sqrt{288} \) as \( 12\sqrt{2} \), so \( 6\sqrt{288}=6\times12\sqrt{2}=72\sqrt{2} \)

\( \sqrt{12}=2\sqrt{3} \), so \( 4\sqrt{12}=8\sqrt{3} \)

So \( h=\frac{72\sqrt{2}-8\sqrt{3}}{3\sqrt{2}}=\frac{72\sqrt{2}}{3\sqrt{2}}-\frac{8\sqrt{3}}{3\sqrt{2}} \)

Simplify \( \frac{72\sqrt{2}}{3\sqrt{2}} = 24 \)

Simplify \( \frac{8\sqrt{3}}{3\sqrt{2}}=\frac{8\sqrt{6}}{6}=\frac{4\sqrt{6}}{3}\approx\frac{4\times2.449}{3}\approx3.265 \)

Wait, this can't be right. Maybe there is a mistake in the problem. Wait, maybe the area is \( 3\sqrt{288}-2\sqrt{18} \)? Let's check. If it is \( 2\sqrt{18} \), then \( \sqrt{18}=3\sqrt{2} \), so \( 2\sqrt{18}=6\sqrt{2} \). Then \( A = 3\times12\sqrt{2}-6\sqrt{2}=36\sqrt{2}-6\sqrt{2}=30\sqrt{2} \). Then \( h=\frac{2\times30\sqrt{2}}{3\sqrt{2}}=\frac{60\sqrt{2}}{3\sqrt{2}} = 20 \). That makes sense. Maybe it's a typo, and the second term is \( 2\sqrt{18} \) instead of \( 2\sqrt{12} \). Alternatively, maybe I misread the problem. Let's re - examine the original problem.

Looking at the image, the area is \( (3\sqrt{288}-2\sqrt{12}) \) and base is \( 3\sqrt{2} \). Wait, let's do the calculation with the given numbers.

First, simplify \( \sqrt{288}=\sqrt{144\times2}=12\sqrt{2} \), so \( 3\sq…

Step 1: Calculate the numerical value of \( \sqrt{6} \)

We know that \( \sqrt{6}\approx2.4495 \)

Step 2: Substitute into the expression for \( h \)

\( h = 24-\frac{4}{3}\times2.4495 \)

First, calculate \( \frac{4}{3}\times2.4495=\frac{9.798}{3}=3.266 \)

Then, \( h = 24 - 3.266=20.734\approx20.73 \) (to the nearest 0.01)

Part (a) Answer: \( \boldsymbol{24-\frac{4\sqrt{6}}{3}} \) (or \( \boldsymbol{\frac{72 - 4\sqrt{6}}{3}} \) or \( \boldsymbol{\frac{4(18-\sqrt{6})}{3}} \))

Part (b) Answer: \( \boldsymbol{20.73} \)

Answer:

Step 1: Calculate the numerical value of \( \sqrt{6} \)

We know that \( \sqrt{6}\approx2.4495 \)

Step 2: Substitute into the expression for \( h \)

\( h = 24-\frac{4}{3}\times2.4495 \)

First, calculate \( \frac{4}{3}\times2.4495=\frac{9.798}{3}=3.266 \)

Then, \( h = 24 - 3.266=20.734\approx20.73 \) (to the nearest 0.01)

Part (a) Answer: \( \boldsymbol{24-\frac{4\sqrt{6}}{3}} \) (or \( \boldsymbol{\frac{72 - 4\sqrt{6}}{3}} \) or \( \boldsymbol{\frac{4(18-\sqrt{6})}{3}} \))

Part (b) Answer: \( \boldsymbol{20.73} \)