Question

1. suppose the top and bottom sides of the rectangle below have a length of ten. suppose the hypotenuse of each triangles has a length of six, and suppose the shorter sides of those same triangles (as visually depicted) each have a length of four. what is the area of the full parallelogram?

image of a parallelogram with a rectangle and two triangles

a) ( 10sqrt{20} )
b) ( 14sqrt{20} )
c) ( 14sqrt{52} )
d) ( 16sqrt{52} )

1. which of the following expressions is equivalent to ( (3x - 4)^2 )?

a) ( 9x^2 + 16 )
b) ( 9x^2 - 16 )
c) ( 9x^2 -24x + 16 )
d) ( 9x^2 - 24x - 16 )

1. a vehicle travels along a straight line, with its position given by ( x(t) = -5t^2 + 12t ). at what time (( t > 0 )) will the vehicle pass through the position ( x = 0 )?

a) 1.8 s
b) 2.4 s
c) 3.6 s
d) it will never pass through ( x = 0 ) for ( t > 0 ).

1. assume a geometric sequence is given by 5, -15, ____, -135. what is the missing term?

a) -75
b) -25
c) 0
d) 45

Explanation:

Question 17

Step1: Find the height of the triangle

Using the Pythagorean theorem \(a^2 + b^2 = c^2\), where \(c = 6\) (hypotenuse) and \(a = 4\) (shorter side). Let the height be \(h\), so \(4^2 + h^2 = 6^2\).
\(16 + h^2 = 36\)
\(h^2 = 36 - 16 = 20\)
\(h=\sqrt{20}\)

Step2: Find the base of the parallelogram

The base of the parallelogram is the length of the rectangle (10) plus the shorter side of the triangle (4) on both sides? Wait, no, looking at the diagram (visually, the parallelogram is made by a rectangle and two triangles). Wait, the top and bottom sides of the rectangle are 10, and each triangle has shorter side 4. Wait, maybe the base of the parallelogram is \(10 + 4=14\)? Wait, no, maybe the two triangles: each has shorter side 4, so total base extension? Wait, the rectangle has length 10, and each triangle has a shorter side (let's say the horizontal side) of 4? Wait, maybe the base of the parallelogram is \(10 + 4 = 14\)? Wait, no, the height of the parallelogram is the height of the triangle, which is \(\sqrt{20}\), and the base of the parallelogram: wait, the rectangle has length 10, and each triangle has a base of 4? Wait, maybe the total base is \(10 + 4=14\)? Wait, no, maybe the two triangles: each has a shorter side (the side adjacent to the rectangle) of 4, so the total base of the parallelogram is \(10 + 4 = 14\)? Wait, the area of a parallelogram is base times height. So base is \(10 + 4 = 14\)? Wait, no, maybe the rectangle is in the middle, and two triangles on the sides. Each triangle has hypotenuse 6, shorter side 4 (let's say the horizontal side). Then the height of the triangle (vertical side) is \(\sqrt{6^2 - 4^2}=\sqrt{36 - 16}=\sqrt{20}\). Then the base of the parallelogram: the rectangle has length 10, and each triangle has a horizontal side of 4? Wait, no, maybe the two triangles: each has a horizontal side of 4, so total base is \(10 + 4 = 14\)? Wait, then area is base times height: \(14\times\sqrt{20}\), which is option B.

Step1: Expand \((3x - 4)^2\) using the formula \((a - b)^2 = a^2 - 2ab + b^2\)

Here, \(a = 3x\) and \(b = 4\).
\((3x)^2 - 2\times(3x)\times4 + 4^2\)

Step2: Calculate each term

\((3x)^2 = 9x^2\), \(2\times(3x)\times4 = 24x\), \(4^2 = 16\).
So, \(9x^2 - 24x + 16\)

Step1: Set \(x(t) = 0\)

We have \(x(t)=-5t^2 + 12t = 0\)

Step2: Solve for \(t\)

Factor out \(t\): \(t(-5t + 12)=0\)
So, \(t = 0\) or \(-5t + 12 = 0\)
Solving \(-5t + 12 = 0\) gives \(5t = 12\) so \(t=\frac{12}{5}=2.4\) seconds (since \(t>0\), we ignore \(t = 0\))

Answer:

B) \(14\sqrt{20}\)

Question 18