Answered: A triangle has side lengths of 5 and 8…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

A triangle has side lengths of 5 and 8 and an area of 16. Show that there are actually two possible triangles that satisfy these conditions by finding the third side of each triangle AND the angle between the side with length 5 and the side with length 8. Round your answers to two decimal places. With the aid of a protractor and a ruler, draw a reasonably accurate diagram of each triangle.

Expert Solution

Step 1

Given that

the two sides of a triangle are 5 and 8

and area of that triangle is 16.

Let sides of the triangle are $a = 5, b = 8, c$

and area is $A = 16$ .

If $s = \frac{a + b + c}{2} = \frac{13 + c}{2}$ , then

Area will be

$A = \sqrt{s (s - a) (s - b) (s - c)}$

$\Rightarrow A = \sqrt{(\frac{13 + c}{2}) (\frac{13 + c}{2} - 5) (\frac{13 + c}{2} - 8) (\frac{13 + c}{2} - c)}$

$\Rightarrow 16 = \sqrt{(\frac{13 + c}{2}) (\frac{3 + c}{2}) (\frac{c - 3}{2}) (\frac{13 - c}{2})}$

$\Rightarrow (\frac{13^{2} - c^{2}}{4}) (\frac{c^{2} - 3^{2}}{4}) = 16^{2}$

$\Rightarrow (13^{2} - c^{2}) (c^{2} - 3^{2}) = 256 \times 16$

$\Rightarrow - c^{4} + (13^{2} + 3^{2}) c^{2} - 169 \times 9 = 4096$

$\Rightarrow c^{4} - 178 c^{2} + 5617 = 0$

$\Rightarrow c^{2} = \frac{178 \pm \sqrt{178^{2} - 4 (5617)}}{2} = \frac{178 \pm \sqrt{9216}}{2} = \frac{178 \pm 96}{2}$

Taking positive sign, we get

$c^{2} = \frac{178 + 96}{2} = \frac{274}{2} = 137$

$\Rightarrow c = \sqrt{137} = 11.70$

and taking negative sign, we get

$c^{2} = \frac{178 - 96}{2} = \frac{82}{2} = 41$

$\Rightarrow c = \sqrt{41} = 6.40$

So two possible triangles are $(5, 8, 11.70)$ and $(5, 8, 6.40)$ which has same area as 16.

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

Martian and hector both dive from a boat to the location shown in the diagram. Find the distance between them. Round to the nearest foot?
Complete the table below, which concerns circles. Complete parts (a) through (d) below Radius Diameter Circumference Area a. 6 cm (Type exact answers, usingz as needed.) b. 24 cm (Type exact answers, using x as needed.) 81x cm (Type exact answers, using x as needed.). d. 36x cm (Type exact answers, using as needed.)

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS