Santiago wants the lateral surfaces of the ramp to be painted red. how many squares feet of the ramp will be painted red?

262 square feet

Step-by-step explanation:

The lateral surface is the side surfaces, except top and bottom.

There are 3 lateral surfaces.

The 2 sides are triangles with length 20 and height 8.5

And

The back is a rectangle with length 12 and height 8.5

We sum up all the 3 surface areas, that will give us the answer.

Area of triangle is (0.5) * length * height

So,

Area of 1 triangle = 0.5 * 20 * 8.5 = 80

Area of 2 triangles = 80 * 2 = 160

Area of rectangle = length * height

so,

Area of back rectangle = 12 * 8.5 = 102

Total Lateral Surface Area = 160 + 102 = 262 square feet

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12

step-by-step explanation: