Sepehr R.

Senior Industrial Engineering Student

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Pre-Calculus

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Question:

Find the unit vector u with the same direction as t=⟨6,-9,-10⟩.

Sepehr R.

Answer:

A unit vector has a magnitude of 1. The magnitude of a vector v =⟨a,b,c⟩ is: | v |= a2+b2+c2 The product of a scalar s and a vector v is the vector s.v. It has magnitude |s|.|v|, where |s| is the absolute value of s and |v| is the magnitude of v. If s is positive, then s.v has the same direction as v. If s is negative, then s.v has the opposite direction of v. Find the magnitude of t=⟨6,-9,-10⟩. Then, divide t by its magnitude to find u. Finally, confirm that u is a unit vector with the same direction as t.

Calculus

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Question:

Find the domain of f(x)=√2−x + √x+5.

Sepehr R.

Answer:

The domain of √2 − x is all real numbers such that 2 − x ≥ 0, that is (−∞, 2]. The domain of √x + 5 is all real numbers such that x+5 ≥ 0, that is [−5, ∞). Therefore the domain of f(x) = √2 − x + √x + 5 is which is [−5, 2]. (−∞, 2] ∩ [−5, ∞)

Physics

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Question:

a 2500 kg car crashes into a tree with a speed of 80 km/h. If fourth of the kinetic energy of the car transferred into heat and that energy is absorbed by the car bumper, by how much is the temperature of the bumper temporarily increased? Specific heat of bumper plastic is about 1800J/(kg.k) and the bumper weight is 19 kg.

Sepehr R.

Answer:

V=80km/1hr*1000m/1km*1hr/3600s= 22.22 m/s K𝐸 = 1/2.𝑚.𝑣^2 K= 1/2* (2500𝑘𝑔)(22.22 )^2= 6.172 *10^5𝐽 Q=1/4*𝐾𝐸 Q=1/4* (6.172 ∗ 10^5𝐽) = 1.54 ∗ 10^5𝐽 Q= 𝑚𝑐∆𝑇 1.54*10?𝐽 = (19𝐾𝑔) ∗ (1800)∆𝑇 ∆𝑇=4.51k

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