Let 
. The tangent plane to the surface at (0, 0, 8) is
The gradient is
so the tangent plane's equation is
The normal vector to the plane at (0, 0, 8) is the same as the gradient of the surface at this point, (1, 1, 1). We can get all points along the line containing this vector by scaling the vector by 
, then ensure it passes through (0, 0, 8) by translating the line so that it does. Then the line has parametric equation
or 
, 
, and 
.
(See the attached plot; the given surface is orange, (0, 0, 8) is the black point, the tangent plane is blue, and the red line is the normal at this point)
Answer:
Question 1: 11%
Question 2: 89%
Question 3: 43%
Question 4: 11%
Step-by-step explanation:
Looking at picture 1, we need to find the crossing point between -1.2 and 0.05. That has 0.1056, which is the same as 10.56%. 10.56% rounds to 11%, so C is our answer.
Picture 2 has the same chart, but we just need to find the inverse, since the inequality sign is flipped. 100 - 10.56 is 89.44%, which rounds to 89%, so D is the answer for Picture 2.
Picture 3 has two tables. 0.73 has 76.73% and -0.41 has 34.09%. Subtract 34.09% from 76.73% to get 42.64% That rounds to 43%, so A is the answer.
Picture 4 essentially has the same expression as Picture 2 (only the sign has switched): P(z ≥ 1.25). The meeting point is 89.44%. Now, subtract that from 100 to get 10.56%, which rounds to 11%. C is our answer for Picture 4.
I hope this helps you! ^w^
8 is to 64 as 2 is to 16
x=16
