Find the area of the helicoid (or spiral ramp) with vector equation r(u, v) = ucos(v) i + usin(v) j + v k, 0 ≤ u ≤ 1, 0 ≤ v ≤ 9π.

The area f the helicoid ramp  is:

∫∫A) | r(u) *r( v) | dudv

The solution is:

A = 10.35×π  square units

r ( u , v ) = u×cosv i +  u×sinv j +v k

To get

r (u ) = δ(r ( u , v ) ) / δu   = [ cosv , sinv , 0 ]

r ( v ) = δ(r ( u , v ) ) / δv  = [ -u×sinv , u×cosv , 1 ]

The vectorial product is:

                                 i                    j              k

r (u ) * r ( v )            cosv             sinv           0

                            -u×sinv           u×cosv       1

r (u ) * r ( v )  =  i × ( sinv - 0 )  - j × ( cosv - 0 ) + k ( u×cos²v + u× sin²v )

r (u ) * r ( v )  = sinv i - cosv j + u k

Now

| r (u ) * r ( v ) |  = √sin²v + cos²v + u²  = √ 1 + u²

Then  

A = ∫₀ (9π) dv    ∫₀¹  √ 1 + u²  du    

 ∫₀¹  √ 1 + u²  du    = 1.15  

A = 1.15 × v |( 0  , 9π )

A = 10.35×π  square units

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the final amount you will pay is $68.04.

the tip is determined by the pre-tax amount, so we calculate the tip based on the original bill, not after we've added sales tax.

to find the tip:

we can divide 54.00 by 100 to get 1% of 54. then we multiply that by 18.

54 / 100 = .54

.54 x 18 = 9.72

tip = 9.72$

to find the tax:

we can divide 54.00 by 100 to get 1% of 54. then we can multiply that by 8 to get 8%.

54 / 100 = .54

.54 x 8 = 4.32

tax= 4.32

now we can add the tip and tax to the bill to get our total.

54 + 9.72 + 4.32 = 68.04

total: $68.04

step-by-step explanation:

x² + 3x + 7 = 0       [ax² + bx + c]

a = 1

b = 3

c = 7

quadratic formula:

    (+/- is this ±)

plug in the numbers into the variables

your answer is a