A parallelogram has sides with lengths of #15 # and #8 #. If the parallelogram's area is #27 #, what is the length of its longest diagonal?

Answer 1

#D~~ 22.866#

Let the first side, #A = 15# Let the second side, #B = 8#

The area is the cross-product of the sides:

#"Area" = A xx B#
#"Area" = |A||B|sin(theta)#
where #theta# is the measure of the smallest angle between #A# and #B#

Substitute the values for the area, A, and B:

#27 = (15)(8)sin(theta)#
#27 = (120)sin(theta)#
#theta = sin^-1(27/120)#
Let #alpha =# the other angle # = pi-theta#
#alpha = pi - sin^-1(27/120)#
Let #D =# the longest diagonal.

We can use the Law of Cosines to compute the length of the longest diagonal:

#D^2 = A^2+B^2-2(A)(B)cos(alpha)#
#D^2 = 15^2+8^2-2(15)(8)cos(pi - sin^-1(27/120))#
#D = sqrt(15^2+8^2-2(15)(8)cos(pi - sin^-1(27/120)))#
#D~~ 22.866#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the length of the longest diagonal of the parallelogram, we can use the formula for the area of a parallelogram, which is given by the product of its base and height. We can set up an equation using this formula, along with the given lengths of the sides:

[ 15 \times h = 27 ]

Solving for the height (( h )), we get:

[ h = \frac{27}{15} = 1.8 ]

Now, we can use the formula for the length of a diagonal in a parallelogram, which is given by:

[ \text{Diagonal} = \sqrt{a^2 + b^2 + 2ab \cos(\theta)} ]

where ( a ) and ( b ) are the lengths of the sides of the parallelogram and ( \theta ) is the angle between those sides. Since we're looking for the longest diagonal, ( \theta = 0 ) and ( \cos(\theta) = 1 ).

So, the length of the longest diagonal (( d )) can be calculated as:

[ d = \sqrt{15^2 + 8^2 + 2(15)(8)(1)} ] [ d = \sqrt{225 + 64 + 240} ] [ d = \sqrt{529} ] [ d = 23 ]

Therefore, the length of the longest diagonal of the parallelogram is ( 23 ) units.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7