Question

1. rosie is comparing $sqrt{7}$ and 3.44444.... she says that $sqrt{7}>3.44444...$ because $sqrt{7}=3.5$. a. what is the correct comparison? b. critique reasoning what mistake did rosie likely make? 15. higher order thinking the length of a rectangle is twice the width. the area of the rectangle is 90 square units. note that you can divide the rectangle into two squares. a. which irrational number represents the length of each side of the squares? b. estimate the length and width of the rectangle. 16. which list shows the numbers in order from least to greatest? a $-4,-\frac{9}{4},\frac{1}{2},3.7,sqrt{5}$ b $-4,-\frac{9}{4},\frac{1}{2},sqrt{5},3.7$

Explanation:

13.

Step1: Find the value of $\sqrt{7}$

We know that $2^2 = 4$ and $3^2=9$, and $7$ is between $4$ and $9$. Using a calculator, $\sqrt{7}\approx2.646$.

Step2: Compare $\sqrt{7}$ and $3.4444\cdots$

Since $2.646<3.4444\cdots$, the correct comparison is $\sqrt{7}<3.4444\cdots$.

Step3: Identify Rosie's mistake

Rosie incorrectly approximated $\sqrt{7}$ as $3.5$. The actual value of $\sqrt{7}\approx2.646$, not $3.5$.

Step1: Let the width of the rectangle be $x$

Since the length of the rectangle is twice the width, the length is $2x$. The area of a rectangle $A = lw$, so $A=(2x)\times x=2x^{2}$. Given that $A = 90$, we have the equation $2x^{2}=90$.

Step2: Solve for $x$

Divide both sides of the equation $2x^{2}=90$ by $2$: $x^{2}=45$. Then $x=\sqrt{45}=3\sqrt{5}$ (we take the positive - value since length cannot be negative). The length of the rectangle is $2x = 6\sqrt{5}$. Since the rectangle can be divided into two squares, the side - length of each square is the width of the rectangle, which is $\sqrt{45}$.

Step3: Estimate the values

We know that $\sqrt{45}$ is between $\sqrt{36}=6$ and $\sqrt{49}=7$. $\sqrt{45}\approx6.7$ (using a calculator). The length of the rectangle $2x = 2\sqrt{45}\approx13.4$.

Step1: Convert $-\frac{9}{4}$ to a decimal

$-\frac{9}{4}=- 2.25$.

Step2: Find the value of $\sqrt{5}$

We know that $2^2 = 4$ and $3^2 = 9$, so $\sqrt{5}\approx2.24$.

Step3: Order the numbers

$-4<-2.25<\frac{1}{2}<2.24<3.7$, so the order is $-4,-\frac{9}{4},\frac{1}{2},\sqrt{5},3.7$.

Answer:

a. $\sqrt{7}<3.4444\cdots$
b. Rosie incorrectly approximated $\sqrt{7}$ as $3.5$.

15.