'(16) Consider the map T : P2(F) →P2(F) defined by T(p(x)) = xp'(x).

(a) Prove that T is linear.

(b) What is null(T)?

(c) Compute the matrix M(T) with respect to the bases B and B' of P2(F), where

B = {1,1 + x,1 + x + x2}, B' = {2−x,x + x2,x2}.

(d) Use the matrix M(T) to write T(7x2 + 12x−1) as a linear combination of the polynomials in the basis B'.

A chorded cycle in a graph is a cycle in the graph with one additional edge connecting two of the cycle vertices. Prove that every graph with minimum degree 3 contains a chorded cycle as a subgraph. (Hint: Consider a longest path in the graph. What does it tell you when a vertex is the end of a longest path? )

When Buying a Lotto Ticket What are my odds? a. Calculate your odds of winning if you spend $1 on an entry. b. Calculate your odds of winning if you spend $50 on an entry. c. Compare these odds to the odds of being in a car accident, plane crash, struck by lightning, or hit by a meterorite. NOTE PLEASE TYPE

How might geometry be used to build a feeling of community? type please

Assume that you have an isosceles right triangle, with legs equal to 6 units, that is rotated around on one of its legs to generate a right circular cone. Find the volume of this cone.

(1) You are estimating the cost of optical sensors based on the radius of the sensors. You decide to calculate the coefficient of determination (R2) as part of determining the goodness of fit of an equation. Using the preliminary calculations below, calculate the R2 and determine its meaning. [Image Description: Summation of (Yi-Ybar)2 = 150000, Summation of (Yhat-Ybar)2 = 125000, Summation of (Y-Yhat)2 = 25000]

16.67% of the variation in the radius is being explained by the cost.

83.33% of the variation in the cost is being explained by the radius.

16.67% of the variation in the cost is being explained by the radius.

83.33% of the variation in the radius is being explained by the cost.

(2) Given the following information, determine the equation for the unit formulation. (Round intermediate calculations to 4 decimal places) [Table Description: Table listing Unit Number (X) and Unit Cost ($Y). Unit 1 = $785.00, Unit 2 = $605.00, Unit 3 =$519.50, Unit 4=$466.27]

Unit Number (X) Unit Cost (Y) $

1 785.00

2 605.00

3 519.50

4 466.27

Y = (785)(X) -0.4941

Y = (785)(X) -0.3758

Y = (785)(X) 0.7707

Y = (785)(X) 0.7 2)

You are estimating the fuel consumption (gallons per mile) for intermediate armored vehicles based on the horsepower of the vehicle. The vehicle for which you are estimating has 4.77 hundred horsepower. Given the following equation, select the correct response from each pair.

Fuel Consumption = - 0.0494 + 0.22 (Horsepower in hundreds)

The dependent variable is Horsepower

The dependent variable is Fuel Consumption The y-intercept is 0.22

The y-intercept is - 0.0494

The estimated fuel consumption for your vehicle is 1.10 gallons per mile

The estimated fuel consumption for your vehicle is 1.00 gallons per mile

(1) Let G be a group and H, K be subgroups of G.
(a) Show that if H is a normal subgroup, then HK = {xy|x ? H, y ? K} is a
subgroup of G.
(b) Show that if H and K are both normal subgroups, then HK is also a normal
subgroup.
(c) Give an example of subgroups H and K such that HK is not a subgroup of G.

Kevin's body metabolizes caffeine at a rate of 14.4% per hour (so the amount of caffeine in Kevin's body decreases by 14.4% each hour).

1. If Kevin consumes a cup of coffee with 86 mg of caffeine in it, how long will it take for Kevin's body to metabolize half of the 86 mg of caffeine?

2. If Kevin consumes an energy drink with 200 mg of caffeine in it, how long will it take for Kevin's body to metabolize half of the 200 mg of caffeine?

3. If Kevin consumes a cup of coffee with cc mg of caffeine in it, how long will it take for Kevin's body to metabolize half of the cc mg of caffeine? (Hint: your answer will be a numerical value.)

How can geometry be used to promote responsible stewardship? Type Please

Given two circles C1 and C2 in the same plane where C1 has a radius of a and C2 has a radius of b and the centers of the C1 and C2 have a distance of c apart. In terms of a,b, and c describe when the intersection of C1 and C2 would be zero points. prove this

State a possible definition for "small triangle" and comment on why such a definition eliminates the counterexamples allowing SAS to be true on the sphere.

Prove that any isometry maps a dinosaur to a dinosaur.

George wants to use square pavers that are 6 inches on each side to completely surround his flower garden. The garden is 12 feet long and 3 feet wide. How many pavers should George buy ? show work

Use the Law of Cosines to solve the triangle. (Let a = 11.9 ft and c = 12.1 ft. Round your answer for b to two decimal places. Round your answers for A and C to the nearest minute.)

one angle is 50 degrees, 30'

A major coffee supplier has warehouses in Seattle and San Jose. The coffee supplier receives orders from coffee retailers in Salt Lake City and Reno. The retailer in Salt Lake City needs 550 pounds of coffee, and the retailer in Reno needs 300 pounds of coffee. The Seattle warehouse has 800 pounds available, and the warehouse in San Jose has 650 pounds available. The cost of shipping from Seattle to Salt Lake City is $2.50 per pound, from Seattle to Reno $ 3 per pound, from San Jose to Salt Lake City $4 per pound, and from San Jose to Reno $2 per pound. Find the number of pounds to be shipped from each warehouse to each retailer to minimize the cost.