Word Problems

Tran paid $$24$ for $32\;[\text{L}]$ of gas. Lilly paid $$27$ for $10\; [\text{gallons}]$ of gas. Who got the better deal?

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In order to properly compare the $2$ payments, we must convert both amounts to the same unit. In this instance, we are converting both amounts to Litres:

$\text{Lily}: (10 \; [\cancel{\text{Gallons}}])\left(\cfrac{3.785411784\; [\text{L}]}{1 \; [\cancel{\text{Gallon}}]}\right)$

$\text{Lily}: 37.85411784\;[\text{L}]$

We can now calculate and compare both payments:

$\text{Lily}: \cfrac{37.85411784 \; [L]}{$27} = 1.4 \; [\text{L}/$]$

$\text{Tran}: \cfrac{32 \; [L]}{$24} = 1.33 \; [\text{L}/$]$

$1.33 < 1.4$

$\text{Tran} < \text{Lily}$

Therefore, we can determine that Tran got a better deal.

The height of Melvin's house is $12\;[\text{m}]$. His friend, Matt, lives in a house that's $15\;[\text{m}]$ tall. If Matt's house casts a shadow that is $16\;[\text{m}]$ long, what is the length of the shadow cast by Melvin's house to the nearest tenth of a metre?

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We can determine the length of the Melvin's houses shadow by using the ratio of both boy's houses heights to their respective shadow lengths and cross-multiplying:

$\cfrac{x}{12\;[\text{m}]} = \cfrac{16\;[\text{m}]}{15\;[\text{m}]}$

$(\cancel{12\;[\text{m}]})\left(\cfrac{x}{\cancel{12\;[\text{m}]}}\right) = (12\;[\text{m}])\left(\cfrac{16\;[\text{m}]}{15\;[\text{m}]}\right)$

$x = \cfrac{192\;[\text{m}^2]}{15 \;[\text{m}]}$

$x = 12.8\;[\text{m}]$

Therefore, we can determine that the length of Melvin's houses shadow is $\textbf{12.8 [m]}$.

The temperature in Jamaica is $80\;[\text{°F}]$ while the temperature in Toronto is $15\;[\text{°C}]$. How much warmer is it in Jamaica than in Toronto? Give your answers in degress Celsius.

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In order to convert Farenheit to Celsius, we must use the following formula:

$\text{°C} = \cfrac{\text{°F} - 32}{1.8}$

Next, we can substitute $80^{\circ}$ and simplify to convert the temperature to $^{\circ}\text{C}$:

$\text{Jamaica}: \cfrac{80° - 32}{1.8}$

$\text{Jamaica}: 27\;[\text{°C}]$

Finally, we can calculate the difference between the 2 countries to determine how much warmer Jamaica is than Canada:

$\text{T} \Delta = \text{Jamiaca} - \text{Canada}$

$\text{T} \Delta = 27\;[°\text{C}] - 15\;[°\text{C}]$

$\text{T} \Delta = 12\;[°\text{C}]$

Therefore, we can determine that Jamaica is roughly $\textbf{12 [°C]}$ warmer than Canada.

Billy's tent is in the shape of a triangular prism.

What is the volume of the tent?
How much material was used to make the tent, including the floor?

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i. First, we can identify the base, $b = 4\;[\text{ft}]$, the height, $h = 3\;[\text{ft}]$ and the length, $l = 7\;[\text{ft}]$.

Next, we can plug the given values into the formula for the Volume of a triangular prism:

$V = \cfrac{1}{2}bhl$

$V = \cfrac{1}{2}(4\;[\text{ft}])(3\;[\text{ft}])(7\;[\text{ft}])$

$V = \cfrac{1}{2}(84\;[\text{ft}^3])$

$V = 42\;[\text{ft}^3]$

Therefore, we can determine that the Volume of the tent is $\textbf{42 [ft³]}$.

ii. We need to identify the total Surface Area of the tent by adding up the individual sides.

First, we can determine the Area of the floor:

$A_{\text{Floor}} = bl$

$A_{\text{Floor}} = (4\;[\text{ft}])(7\;[\text{ft}])$

$A_{\text{Floor}} = 28\;[\text{ft}^2]$

Next, we can determine the Areas of the triangular faces:

$A_{\text{Triangular Faces}} = 2 \cdot \cfrac{1}{2} bh$
$A_{\text{Triangular Faces}} = \cancel{2} \cdot \cancel{\cfrac{1}{2}}(3\;[\text{ft}])(4\;[\text{ft}])$
$A_{\text{Triangular Faces}} = 12\;[\text{ft}^2]$

Finally, we can determine the Areas of the rectangular sides. We first need to calculate the hypoteneuse to determine their widths, $w$:

$w^2 = a^2 + b^2$

$w^2 = 3^2 + 2^2$

$\sqrt{w^2} = \sqrt{14}$

$w = 3.74\;[\text{ft}]$

We can now calculate the Areas of the rectangular sides:

$A_{\text{Rectangular Sides}} = 2lw$

$A_{\text{Rectangular Sides}} = 2(7\;[\text{ft}])(3.74\;[\text{ft}])$

$A_{\text{Rectangular Sides}} = 52.36\;[\text{ft}^2]$

Finallly, we can calculate the sum of all the individual Areas to determine the total Surface Area:

$\text{SA} = A_{\text{Floor}} + A_{\text{Triangular Faces}} + A_{\text{Rectangular Sides}}$

$\text{SA} = 28\;[\text{ft}^2] + 12\;[\text{ft}^2] + 52.36\;[\text{ft}^2]$

$\text{SA} = 92.36\;[\text{ft}^2]$

Therefore, we can determine that we need $\boldsymbol{92.36 \; [\textbf{ft}^2]}$ of wood to make the tent.

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