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As in the text, consider biodiesel and petrodiesel mixtures that are 30% biodiesel and 70% petrodiesel. Let B stand for an unspecified number of liters of biodiesel, which can vary, and let P stand for the corresponding number of liters of petrodiesel in such mixtures. Reason about quantities and our definition of multiplication, and use a math drawing to explain the equation
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- 6. [10] Evaluate (2 (2+2y) ds where C is the upper half-circle centered at the origin connecting the point (2,0) to the point (-2,0). 7. [10] Show that the vector field F(x, y, z) = is conservative and integrate it along the curve П C(t) = si sin t, nt, cost,t), te [0,1] 8. [10] Use Stokes' Theorem to compute the integral curl F.dS, where F(x, y, z) = xzi+yxj+xy k I cur and S is the part of the sphere x² + y²+z² = 9 that lies inside the cylinder x2 + y² above the xy-plane. 9. [10] Use Green's theorem to evaluate So √1+x3 dx+2xy dy where C is the triangle with vertices (0,0), (1, 0) and (1, 3). 10. [10] Evaluate the surface integral (x²z + y²z) dS where S is the hemisphere x² + y²+2² = 4, z> 0. = 1 and ☐arrow_forwardPart 1: A linear electrical load draws 1₁ A at a 0.72 lagging power factor. See the table to find ½ for your student ID. When a capacitor is connected, the line current dropped to 122 A and the power factor improved to 0.98 lagging. Supply frequency is 50 Hz. a. Let the current drawn from the source before and after introduction of the capacitor be 1₁ and I₂ respectively. Take the source voltage as the reference and express 11 and 12 as vector quantities in polar form. b. Obtain the capacitor current, Ic = 12 − I₁, graphically as well as using complex number manipulation. Compare the results. c. Express the waveforms of the source current before (į (t)) and after (i2(t)) introduction of the capacitor in the form Im sin(2лft + 0). Hand sketch them on the same graph. Clearly label your plots. d. Analytically solve i̟2(t) − i₁ (t) using the theories of trigonometry to obtain the capacitor current in the form, ic(t) = Icm sin(2πft + 0c). Compare the result with the result in Part b.arrow_forward= x³, y = 8, x = 0. Let R be the region bounded by the curves y = x³ 1. Sketch the region and find the area. Write your answer in simplest fractional form. 2. Sketch the solid you obtain by rotating the region R about the x-axis. 3. Find the volume of the solid obtained by rotating the region R about the x-axis using the disk/washer method. Write the formula you are using. Write your answer in terms of π. Draw the approximating rectangle that you rotate. 4. Find the volume of the solid obtained by rotating the region R about the x-axis using the shell method. Write the formula you are using. Write your answer in terms of π. Draw the approximating rectangle that you rotate. 5. Which method did you find easier and why? [There is no wrong answer for what you find easier, but explain.] 6. Sketch the solid you obtain by rotating the region R about the y-axis. 7. Find the volume of the solid obtained by rotating the region R about the y-axis using the disk/washer method. Write the formula…arrow_forward
- Go to page 82 for the geometry problem. Use the formula for the area of a triangle to compute the area given the base and height. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxp RyejKEMg 1t2q15dbpVLCS/view? usp=sharing] Provide a step-by-step solution.arrow_forwardRefer to page 79 of the shared document for the algebra problem. Use basic algebraic rules to simplify the given expression. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxp RyejKEMg1t2q15dbpVLCS/view? usp=sharing] Provide all steps clearly.arrow_forward#7 Using implicit differentiation, find the equation of the tangent line to the given curve at the given point: a) 3x2y2-3y-17=5x+14 at (1,-3) b) y2-7xy+x-2x=9 at (0,3)arrow_forward
- Please calculate the shaded areaarrow_forwardAn investigator analyzed the leading digits from 797 checks issued by seven suspect companies. The frequencies were found to be 0, 19, 2, 50, 361, 309, 10, 22, and 24, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the observed frequencies are substantially different from the frequencies expected with Benford's law shown below, the check amounts appear to result from fraud. Use a 0.10 significance level to test for goodness-of-fit with Benford's law. Does it appear that the checks are the result of fraud? Leading Digit Actual Frequency Benford's Law: Distribution of Leading Digits 1 2 3 4 5 6 7 8 9 0 19 2 50 361 309 10 22 24 30.1% 17.6% 12.5% 9.7% 7.9% 6.7% 5.8% 5.1% 4.6% Determine the null and alternative hypotheses. Ho The leading digits are from a population that conforms to Benford's law. H₁: At least one leading digit has a frequency that does not conform to Benford's law. Calculate the test statistic, x². x² = (Round to three…arrow_forward3. Solve the Heat Equation with Initial and Boundary Conditions Turn to page 71 for the heat equation problem. Solve the partial differential equation using Fourier series or another suitable method, given the initial and boundary conditions. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Provide all derivations and intermediate steps.arrow_forward
- Name: Tay Jones Level Two Date: Algebra 3 Unit 3: Functions and Equations Practice Assessment Class: #7-OneNote 1. The function f(x) = x² is transformed in the following functions. List the vertex for each function, circle whether the function opens up or down, and why. All three parts must be correct to receive Level 2 points. You can receive points for a, b, and c. a) g(x) = -2(x+5)² Vertex: Opens Up Opens Down Why? ais negative -2 Vertex: b) g(x) = (x + 2)² - 3 c) g(x) = -4(x + 2)² + 2 Opens Up Opens Down Vertex: Opens Up Opens Down Why? 4 Ca is negative) Why? his positive 2. The graph of the function f(x) is shown below. Find the domain, range, and end behavior. Then list the values of x for which the function values are increasing and decreasing. f(x) Domain: End Behavior: As x → ∞o, f(x) -> -6 As x, f(x) -> Range: Where is it Increasing? (002] Where is it Decreasing? (1,00)arrow_forwardFor the distribution drawn here, identify the mean, median, and mode. Question content area bottom Part 1 A. Aequalsmode, Bequalsmedian, Cequalsmean B. Aequalsmode, Bequalsmean, Cequalsmedian C. Aequalsmedian, Bequalsmode, Cequalsmean D. Aequalsmean, Bequalsmode, Cequalsmedianarrow_forwardQ3: Define the linear functional J: H₁(2) R by ¡(v) = a(v, v) - L(v) Л Let u be the unique weak solution to a(u,v) = L(v) in H(2) and suppose that a(...) is a symmetric bilinear form on H(2) prove that 1- u is minimizer. 2- u is unique. 3- The minimizer J(u) can be rewritten under 1(u) = u Au-ub, algebraic form 1 2 Where A, b are repictively the stiffence matrix and the load vector Q4: A) Answer 1- show that the solution to -Au = f in A, u = 0 on a satisfies the stability Vullfll and show that ||V(u u)||||||2 - ||vu||2 2- Prove that Where lu-ul Chuz - !ull = a(u, u) = Vu. Vu dx + fu. uds B) Consider the bilinea forta Л a(u, v) = (Au, Av) (Vu, Vv + (Vu, v) + (u,v) Show that a(u, v) continues and V- elliptic on H(2)arrow_forward
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